Book of Abstracts - IRIT

Revisiting the bookkeeping technique in HOAS-based encodings
Ciaffaglione and Scagnetto
where, of course, we denote by Ti the Coq encoding of T i . So doing, the representation of a
typing environment is not confined to a list, but it is “global”, in the sense that every variable
declaration is always accessible. This implies that we can encode the subtyping relation as an
inductive predicate subTp: Tp -> Tp -> Prop, without carrying around the environment.
Obviously, there are some drawbacks to deal with. First, being envTp an open predicate, we
must enforce by some Axioms the usual good formation conditions about contexts. Secondly,
a more subtle issue must be handled; for instance, let us consider the informal narrowing
statement:
Γ, X (notin_ho Y M) -> (notin_ho Y N) ->
(envTp Y P) -> (subTp (M Y) (N Y)).
According to our experiments with these kinds of encodings, the implicit representation of
contexts Γ works well with the ToC, because we can derive specialized induction principles
which are more powerful (from a practical point of view) than Coq’s built-in induction tactic.
In fact, the trick for System F