pith. machine review for the scientific record. sign in
theorem proved tactic proof

weighted_Jlog_average

show as:
view Lean formalization →

No prose has been written for this declaration yet. The Lean source and graph data below render without it.

generate prose now

formal statement (Lean)

 145private theorem weighted_Jlog_average {N : ℕ} (c : Configuration N) :
 146    (∑ i ∈ (Finset.univ : Finset (Fin N)), (1 / (N : ℝ)) * Jlog (Real.log (c.entries i))) =
 147      (1 / (N : ℝ)) * total_defect c := by

proof body

Tactic-mode proof.

 148  calc
 149    (∑ i ∈ (Finset.univ : Finset (Fin N)), (1 / (N : ℝ)) * Jlog (Real.log (c.entries i)))
 150        = ∑ i ∈ (Finset.univ : Finset (Fin N)), (1 / (N : ℝ)) * defect (c.entries i) := by
 151            apply Finset.sum_congr rfl
 152            intro i _
 153            unfold Jlog defect J Jcost
 154            rw [Real.exp_log (c.entries_pos i)]
 155    _ = (1 / (N : ℝ)) * total_defect c := by
 156          unfold total_defect
 157          rw [← Finset.mul_sum]
 158
 159/-- Jensen lower bound: fixed log-charge implies a defect lower bound. -/

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (7)

Lean names referenced from this declaration's body.