This talk discusses current extensions on the theory algebra from the NASA/PVSlibrary on formal developments for the Prototype Verification System (PVS). It discusses the approach to formalizing theorems of the ring theory and how they are applied to infer properties of specific algebraic structures. As cases of study, we will present recent formalizations on the theories of Euclidean Domains and Quaternions. Moreover, we will show how a general verification of Euclid’s division algorithm can be specialized to verify this algorithm for specific Euclidean Domains, and how the abstract theory of Quaternions can be parameterized to deal with the structure of Hamilton’s Quaternions.

A mechanical proof of the Knuth–Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet’s well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth–Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.