Abstract
We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. types endowed with an arbitrary equivalence relation, as carriers for algebras. In this way it is possible to define the quotient of an algebra by a congruence. Standard constructions over algebras are defined and their basic properties are proved formally. To overcome the problem of defining term algebras in a uniform way, we use types of trees that generalize wellorderings. Our implementation gives tools to define new algebraic structures, to manipulate them and to prove their properties.
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Capretta, V. (1999). Universal Algebra in Type Theory. In: Bertot, Y., Dowek, G., Théry, L., Hirschowitz, A., Paulin, C. (eds) Theorem Proving in Higher Order Logics. TPHOLs 1999. Lecture Notes in Computer Science, vol 1690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48256-3_10
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DOI: https://doi.org/10.1007/3-540-48256-3_10
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