Automated Verification of Fundamental Algebraic Laws

Algebraic laws of functions in mathematics -- such as commutativity, associativity, and idempotence
-- are often used as the basis to derive more sophisticated properties of complex
mathematical structures and are heavily used in abstract computational thinking.
Algebraic laws of functions in coding, however, are rarely considered.
Yet, they are essential. For example, commutativity and associativity are crucial to ensure correctness of 
a variety of software systems in numerous domains,
such as compiler optimization, big data processing, data flow processing,
machine learning or distributed algorithms and data structures.
Still, most programming languages lack built-in mechanisms
to enforce and verify that operations adhere to such properties.

In this artifact, we propose a verifier specialized on a set of fundamental algebraic laws that
ensures that such laws hold in application code.
The verifier can conjecture auxiliary properties and can reason about both equalities
and inequalities of expressions, which is crucial to prove a given property
when other competitors do not succeed.
We implement these ideas in the Propel verifier.
Our evaluation against five state-of-the-art verifiers on a total of 142 instances of algebraic properties
shows that Propel is able to automatically deduce algebraic properties
in different domains that rely on such properties for correctness,
even in cases where competitors fail to verify the same properties or time out.