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a2ee19a | 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 57 58 | import OAI.Probability.SATComputability.MeanSurvival
namespace OAI
namespace FixedClauseThreshold.Computability
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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