| import OAI.Probability.SATComputability.MeanSurvival |
|
|
| namespace OAI |
|
|
| namespace FixedClauseThreshold.Computability |
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|
| open DilutedSpinGlass _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory |
| open scoped Classical NNReal |
|
|
| theorem candidateStep_minimumDeletions_le {n : ℕ} [NeZero n] (m : ℕ) : |
| (candidateStep (n := n) 3)^[m] (fun U => (minimumDeletions U : ℝ)) Finset.univ ≤ |
| relaxedFixedMinimum n m := by |
| rw [candidateStep_iterate, finite_uniform_expect_eq_mean] |
| let e := Equiv.piCongrRight (fun _ : Fin m => relaxedClauseEquiv n) |
| rw [← uniformMean_equiv e] |
| apply uniformMean_mono |
| intro cs |
| exact_mod_cast candidate_minimumDeletions_le cs |
|
|
| theorem candidateBlock_minimumDeletions_le (n : ℕ) [NeZero n] (a : ℝ≥0) : |
| (candidateBlock (n := n) (a*n) 3 Finset.univ).expect |
| (fun U => (minimumDeletions U : ℝ)) ≤ relaxedPoissonMinimum n a := by |
| rw [candidateBlock_expect, relaxedPoissonMinimum] |
| apply integral_mono |
| · simpa only [candidateFixedBlock_expect] using |
| poissonMixture_integrable_expect (a*n) |
| (fun m => candidateFixedBlock (n := n) 3 m Finset.univ) |
| (fun U => (minimumDeletions U : ℝ)) |
| · apply (poisson_integrable_count (a*n)).mono' |
| (measurable_of_countable _).aestronglyMeasurable |
| exact ae_of_all _ (fun m => by |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (relaxedFixedMinimum_nonneg n m)] using |
| relaxedFixedMinimum_le n m) |
| · exact candidateStep_minimumDeletions_le |
|
|
| theorem mean_lifetime_energy_lower {n r t : ℕ} [NeZero n] |
| (hr : r ≤ n+1) (ht : 0 < t) (htM : t ≤ 20*n) (a : ℝ≥0) |
| (hta : (t : ℝ≥0) ≤ a*n) : |
| (r : ℝ)*(1- |
| (FiniteLaw.pi (fun _ : Fin (20*n) => candidateBlock (n := n) 1 3 Finset.univ)).expect |
| (maskLifetime (20*n) (deletionBudgetMask n r)) / t) ≤ |
| relaxedPoissonMinimum n a := by |
| have htR : (0 : ℝ) < t := by exact_mod_cast ht |
| have hs := survival_le_mean_lifetime htM (deletionBudgetMask n r) |
| have hp := candidateBlock_survival_antitone 3 (deletionBudgetMask n r) hta |
| have hq : (candidateBlock (n := n) t 3 (deletionBudgetMask n r)).expect candidateAlive ≤ |
| (FiniteLaw.pi (fun _ : Fin (20*n) => candidateBlock (n := n) 1 3 Finset.univ)).expect |
| (maskLifetime (20*n) (deletionBudgetMask n r)) / t := |
| (le_div_iff₀ htR).mpr (by nlinarith only [hs]) |
| have hb := block_minimumDeletions_lower hr (a*n) |
| have hm := candidateBlock_minimumDeletions_le n a |
| exact (mul_le_mul_of_nonneg_left (sub_le_sub_left (hp.trans hq) 1) |
| (Nat.cast_nonneg r)).trans (hb.trans hm) |
|
|
| end FixedClauseThreshold.Computability |
|
|
| end OAI |
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|
