Extensional equality preservation and verified generic programming

TL;DR

This paper investigates how preserving extensional equality in functors and monads within intensional type theories can enable simpler, verified generic proofs and applications in dynamical systems and control theory.

Contribution

It introduces a minimalist approach to extensional equality in intensional type theories, facilitating verified generic programming without complex setoid constructions.

Findings

Enables simple generic proofs of monadic laws

Provides verified results in dynamical systems and control theory

Avoids tedious code duplication and ad-hoc proofs

Abstract

In verified generic programming, one cannot exploit the structure of concrete data types but has to rely on well chosen sets of specifications or abstract data types (ADTs). Functors and monads are at the core of many applications of functional programming. This raises the question of what useful ADTs for verified functors and monads could look like. The functorial map of many important monads preserves extensional equality. For instance, if $f, g : A \rightarrow B$ are extensionally equal, that is, $\forall x \in A, \ f \ x = g \ x$, then $map \ f : List \ A \rightarrow List \ B$ and $map \ g$ are also extensionally equal. This suggests that preservation of extensional equality could be a useful principle in verified generic programming. We explore this possibility with a minimalist approach: we deal with (the lack of) extensional equality in Martin-L\"of's intensional type theories without extending the theories or using full-fledged setoids. Perhaps surprisingly, this minimal approach turns out to be extremely useful. It allows one to derive simple generic proofs of monadic laws but also verified, generic results in dynamical systems and control theory. In turn, these results avoid tedious code duplication and ad-hoc proofs. Thus, our work is a contribution towards pragmatic, verified generic programming.