This is a formalisation in Agda of some of the material presented by Bob Harper at OPLSS 2011. It is developed by James Chapman and Andri Saar with the aid of some useful discussion about with Taus Brock-Nannestad and Filip Sieczkowski. https://github.com/finrod/System-T for Filip's formalisation in Coq. Terms.agda ========== This file contains the term syntax of system T. It is indexed by contexts and types in is hence guaranteed to be well-scoped and well-typed. Variables are represented as de Bruijn indices. You can read them as natural numbers but are guaranteed not to point out of scope and also carry their type. Renamings.agda ============== This file contains the renaming operations on terms and associated lemmas. We need this as we use it in our definition of substitution. Renamings are represented as functions from sets variables to sets of variables. The benefit of this functional representation (as opposed to a first order one) is that we get some stuff for free, we don't need to prove left and right identity and associativity of renamings as they hold definitionally. The downside is that when we do need to reason about renamings we need to use extensionality. Substitutions.agda ================== This file contains the substitution operations and associated lemmas. Like renamings we represent substitutions as functions, this time from variables to terms. The renaming and substitution machinery was developed in the paper "Relative monads formalised". See my webpage for a draft http://www.cs.ioc.ee/~james/Publications.html. The particular representation was chosen as it corresponds exactly to 'bind' of the relative monad given by the typed lambda terms. The lemmas proven are those necessary to check the (relative) monad laws. Substitutions are Kleisli maps and the EM algebras are evaluators (models) which I think is pretty cool. Equality.agda ============= This file contains one utility function (congruence of ternary functions) that I couldn't find in the standard library that we need and also postulates extensionality. Reducibility.agda ================= This file contains the guts of the proof. It is very close to what Bob presented I think. In fact, because we are working in Agda we can simplify things a bit. We don't need to worry about the properties of the reducibility predicate on natural numbers. It is an inductive definition so we get what we need for free. We also avoid the predicate P and its closure, probably for the same reason. James