Formalisation of Polynomials in Cubical Type Theory Using Cubical Agda

Abstract: We formalise polynomials over commutative rings in cubical type theory using Cubical Agda as proof assistant. On the basis of a formalisation of polynomials as number sequences with only a finite number of non-zero values, we use higher inductive types to formulate a list-based definition using two point constructors and two path constructors. The combinatorial explosion in proofs leads us to a redefinition: One of the path constructors is discarded, and instead we formulate a separate function-based definition. We prove equivalence of these distinct definitions, and use the function-based definition to provide a witness for the discarded path constructor. The list-based definition is then used in combination with this witness to prove that the resulting structure is itself a commutative ring.