Alpha-Structural Induction and Recursion for the Lambda Calculus in Constructive Type Theory

Formalisation in Constructive Type Theory of Barendregt's Variable Convention for Generic Structures with Binders

“…Note that the swapping is performed only over the sort of variables bound by this binder position, leaving any other sort of variables unchanged. We are able to prove that this is an equivalence relation, and also that it is preserved under name-swapping in a similar way as done in our previous work [7].…”

“…First, we define an auxiliary relation ∼αF, which is inductively defined introducing an auxiliary functor G, used to traverse the functor F structure. For the interesting binder case, we follow an idea similar to the one used in [7], that is, we define that two abstractions are α-equivalent if there exists some list of variables xs, such that for any given variable z not in xs, the result of swapping the corresponding binders with z in the abstraction bodies is α-equivalent. Note that the swapping is performed only over the sort of variables bound by this binder position, leaving any other sort of variables unchanged.…”

“…This iteration principle first finds a fresh term for a given context c, and then directly applies the fold operation over it. We developed this iteration principle following a different approach from the one taken in our previous work [7], where we renamed the binders during the fold traverse. Instead, we chose to separate these two stages in order to reuse the previously defined fold operation and its properties.…”

Unificação, confluência e tipos com interseção para sistemas de reescrita nominal

“…An induction principle was implemented in order to prove properties about well-formed terms without mentioning indices. [Copello et al, 2016] also presents an α-structural induction principle in Agda and proves the Substitution Lemma using such an inductive scheme, but it criticises the use of higher-order features in [Urban, 2008] and the indices in [Aydemir et al, 2007] to represent bindings. Instead, the authors claim to use a method similar to the Barandregt's Variable Convention, where a bound name is suposed to be chosen different from a given list of names.…”

“…To the best of our knowledge, our type system is the first one that works with intersections in the context of nominal terms. It is based on [van Bakel, 1995, van Bakel andFernández, 1997] with respect to the intersection type features, and on [Fairweather, 2014, Fairweather andFernández, 2016] regarding the nominal restrictions to obtain properties such as subject reduction.…”