trop_sInf_image — Mathlib

English

There is an instance asserting that Tropical (WithTop R) has no zero divisors when R is AddCommMonoid.

Русский

Существует экземпляр, подтверждающий отсутствие нулевых делителей в Tropical(WithTop R) при R аддитивном моноиде.

LaTeX

$$$\mathrm{NoZeroDivisors}(\mathrm{Tropical}(\mathrm{WithTop}(R)))$$$

Lean4

theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) :
    trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) :=
  by
  rcases s.eq_empty_or_nonempty with (rfl | h)
  · simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, trop_top]
  rw [← inf'_eq_csInf_image _ h, inf'_eq_inf, s.trop_inf]

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