File size: 2,197 Bytes
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.DeletionMean
import OAI.Probability.SATComputability.DyadicBlocks
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass MeasureTheory
open scoped BigOperators
theorem sixth_fifth_power_le_sum (d : ℕ) :
(d : ℝ)^(6/5 : ℝ) ≤
(6/5 : ℝ) * ∑ j ∈ Finset.range d, ((j : ℝ)+1)^(1/5 : ℝ) := by
have hm : MonotoneOn (fun x : ℝ => x^(1/5 : ℝ)) (Set.Icc 0 (0+(d : ℝ))) := by
intro x hx y _ hxy
exact Real.rpow_le_rpow hx.1 hxy (by norm_num)
have h := hm.integral_le_sum
simp only [zero_add, Nat.cast_add, Nat.cast_one] at h
rw [integral_rpow (Or.inl (by norm_num : (-1 : ℝ) < 1/5))] at h
norm_num at h
nlinarith
theorem finite_fractional_moment {Ω : Type*} [Fintype Ω] (P : FiniteLaw Ω)
(f : Ω → ℝ) (hf : ∀ ω, 0 ≤ f ω) :
P.expect (fun ω => (f ω)^(1/5 : ℝ)) ≤ (P.expect f)^(1/5 : ℝ) := by
exact (Real.concaveOn_rpow (by norm_num : (0 : ℝ) ≤ 1/5)
(by norm_num : (1/5 : ℝ) ≤ 1)).le_map_sum
(fun ω _ => P.nonneg ω) P.total (fun ω _ => hf ω)
theorem reverse_interval_sum (A B : ℕ) (hAB : A ≤ B) (f : ℕ → ℝ) :
(∑ t ∈ Finset.Ico A B, f (B-t)) =
∑ j ∈ Finset.range (B-A), f (j+1) := by
rw [Finset.sum_Ico_eq_sum_range, ← Finset.sum_range_reflect]
apply Finset.sum_congr rfl
intro j hj
congr 1
have hj' := Finset.mem_range.mp hj
omega
theorem discrete_delay_moment {A B M : ℕ} (hAB : A ≤ B) (hBM : B ≤ M) :
((B-A : ℕ) : ℝ)^(6/5 : ℝ) ≤ (6/5 : ℝ) *
∑ t ∈ Finset.range M,
if A ≤ t ∧ t < B then ((B-t : ℕ) : ℝ)^(1/5 : ℝ) else 0 := by
have he : (∑ t ∈ Finset.range M,
if A ≤ t ∧ t < B then ((B-t : ℕ) : ℝ)^(1/5 : ℝ) else 0) =
∑ t ∈ Finset.Ico A B, ((B-t : ℕ) : ℝ)^(1/5 : ℝ) := by
rw [← Finset.sum_filter]
congr 1
ext t
simp only [Finset.mem_filter, Finset.mem_range, Finset.mem_Ico]
omega
rw [he, reverse_interval_sum A B hAB (fun j => (j : ℝ)^(1/5 : ℝ))]
simpa only [Nat.cast_add, Nat.cast_one] using sixth_fifth_power_le_sum (B-A)
end FixedClauseThreshold.Computability
end OAI
|