The Foldl Operator as a Coequalizer Using Coq

Abstract

In the present work a Coq based approach is taken to characterize the foldl using a functorial structure from which an inductive type is determined. With μ _F being an initial F–algebra and (B,θ) another F–algebra, two F–algebras with support B×μ _F are constructed and then coequalized. This coequalization morphism allows the definition of foldl structurally.

After examining some significant examples we propose the following methodology to define a foldl operator. Let F be a polynomial endofunctor and (μ _F ,in _F ) its initial algebra. We define two F–algebras with support B×μ _F , and h₁,h₂:F(B×μ _F ) →B ×μ _F constructed such that in one of them the argument of the initial type is syntactically (structurally) lower than that in the other. Then, ${\mathit{foldl}}:B\times\mu_F \rightarrow B$ can be defined as a specific morphism that coequalizes them $(h_1;{\mathit{foldl}}=h_2;{\mathit{foldl}}).$

For an initial F–algebra with distinguished element (as in the case of lists), foldl is a coequalizer of h₁ and h₂.

The proofs are performed using the Coq proof system. In this context, constructive approach stress that existence is constructive, therefore with a computational content.

Detailed structure for proofs in Coq is included but interactive code is omited. See the section 4 for a repository with the complete source code.

Partially supported by Xunta de Galicia PGIDIT07TIC005105PR and MEC TIN2005-08986.