of integers $\mathbb Z$ and any skew field were proven proper in [@beal.lombardy.sakarovitch:2006 Theorem 1]. Here, all zig-zag were spans, i.e., had a single intermediate node with outgoing arrows.

-   Noetherian semirings were proven proper in [@esik.maletti:2010 Theorem 4.2], commutative rings also in [@esik.maletti:2010 Corollary 4.4], and finite semirings as well in [@esik.maletti:2010 Corollary 4.5], all with a zig-zag being a span. Moreover, the tropical semiring is not proper, as proven in [@esik.maletti:2010 Theorem 5.4].

Having properness of a semiring, together with the property of the semiring being finitely and effectively presentable, yields decidability of the equivalence problem (decidability of trace equivalence) for weighted automata.

In this paper, motivated by the wish to prove properness of a certain functor ${{\ensuremath{\widehat F}}}$ on convex algebras used for axiomatizing trace semantics of probabilistic systems in [@silva.sokolova:2011], as well as by the open questions stated in [@milius:2017 Example 3.19], we provide a framework for proving properness. We instantiate this framework on known cases like Noetherian semirings and $\mathbb N$ (with a zig-zag that is a span), and further prove new results of properness:

-   The semirings $\mathbb Q_+$ and $\mathbb R_+$ of non-negative rationals and reals, respectively, are proper. The shape of the 