Monoid Theory in Alonzo: A Little Theories Formalization in Simple Type Theory

TL;DR

Monoid theory is formalized in Alonzo using the little theories method, organizing concepts as a modular development graph and transporting them via theory morphisms between MON, COM-MON, MON-ACT, and related constructs. The approach supports undefined expressions and partial functions, enabling natural representations of algebraic structures such as $(m,\cdot,e)$, its opposite $(m,\cdot^{\mathrm{op}},e)$, and set and transformation monoids, while prioritizing communication over full machine-checked proofs. The work further demonstrates transport to strings via MON-over-COF to STR and to real-number arithmetic via MON-over-COF, culminating in a strings monoid and related structures (e.g., ${\mathsf{MONOID}}(\mathsf{str}_{\{ R\rightarrow A\}},\mathsf{cat},\epsilon)$). The resulting development graph $G_{\rm mon}$ showcases high modularity, reuse, and scalability, and points toward extending the framework to related domains such as group theory and broader formal libraries.

Abstract

Alonzo is a practice-oriented classical higher-order version of predicate logic that extends first-order logic and that admits undefined expressions. Named in honor of Alonzo Church, Alonzo is based on Church's type theory, Church's formulation of simple type theory. The little theories method is a method for formalizing mathematical knowledge as a theory graph consisting of theories as nodes and theory morphisms as directed edges. The development of a mathematical topic is done in the "little theory" in the theory graph that has the most convenient level of abstraction and the most convenient vocabulary, and then the definitions and theorems produced in the development are transported, as needed, to other theories via the theory morphisms in the theory graph. The purpose of this paper is to illustrate how a body of mathematical knowledge can be formalized in Alonzo using the little theories method. This is done by formalizing monoid theory -- the body of mathematical knowledge about monoids -- in Alonzo. Instead of using the standard approach to formal mathematics in which mathematics is done with the help of a proof assistant and all details are formally proved and mechanically checked, we employ an alternative approach in which everything is done within a formal logic but proofs are not required to be fully formal. The standard approach focuses on certification, while this alternative approach focuses on communication and accessibility.

Figures (1)

• Figure 1: The Monoid Theory Development Graph

Theorems & Definitions (2)

• Lemma B.1: Universal Sets
• Lemma B.2