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

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.