We introduce the concept of compact quantitative equational theory. A quantitative equational theory is defined to be compact if all its consequences are derivable by means of finite proofs. We prove that the theory of interpolative barycentric (also known as convex) quantitative algebras of Mardare et. al. is compact. This serves as a paradigmatic example, used to obtain other compact quantitative equational theories of convex algebras, each axiomatizing some distance on finitely supported probability distributions.

翻译：我们引入了紧致定量等式理论的概念。一个定量等式理论被定义为紧致的，如果其所有推论均可通过有限证明推导得出。我们证明了Mardare等人提出的插值重心（也称为凸）定量代数的理论是紧致的。这作为一个范例，用于获得凸代数的其他紧致定量等式理论，每个理论均公理化有限支撑概率分布上的某种距离。