sum_ofWithBot — Mathlib

English

A variant of monotonicity for ofWithBot under different indexing assumptions.

Русский

Вариант монотонности for ofWithBot при иных предположениях об индексировании.

LaTeX

$$$ (\\text{conditions}) \\Rightarrow ofWithBot boxes_1 le ofWithBot boxes_2 $$$

Lean4

theorem sum_ofWithBot {M : Type*} [AddCommMonoid M] (boxes : Finset (WithBot (Box ι)))
    (le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I)
    (pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) (f : Box ι → M) :
    (∑ J ∈ (ofWithBot boxes le_of_mem pairwise_disjoint).boxes, f J) = ∑ J ∈ boxes, Option.elim' 0 f J :=
  Finset.sum_eraseNone _ _

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