linearGrowthSup_sup — Mathlib

English

If ι is finite, linearGrowthSup (iSup i, u i) equals iSup i, linearGrowthSup (u i).

Русский

Если ι конечен, то верхний рост по iSup равен верхнему росту суммы по каждому i.

LaTeX

$$$\operatorname{linearGrowthSup}\bigl(\iSup i, u(i)\bigr) = \iSup i, \operatorname{linearGrowthSup}(u(i))$$$

Lean4

theorem linearGrowthSup_sup : linearGrowthSup (u ⊔ v) = max (linearGrowthSup u) (linearGrowthSup v) :=
  by
  rw [linearGrowthSup, linearGrowthSup, linearGrowthSup, ← limsup_max]
  refine limsup_congr (Eventually.of_forall fun n ↦ ?_)
  exact (monotone_div_right_of_nonneg n.cast_nonneg').map_max

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