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Free keywords:
Dependent Type Theory; Verified generic programming; Extensional Equality
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 → B are extensionally equal, that
is, ∀x ∈ A, f x = g x, then map f : List A → 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-Löf’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.
