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Lambda Calculus in Coq

A from-scratch formalization of untyped lambda calculus with de Bruijn indices in Coq+SSReflect.

With this project, I try to brush up my Coq knowledge and learn more about how de Bruijn indices are implemented. Last time, in my formalization of a basic ontology language I learned the hard way that named variables are perhaps not that easy as on pen & paper. Hence, this time I am trying de Bruijn indices from the start.

In utlc.v we start off pretty easy with

(* Untyped LC (UTLC) *)
Inductive LC :=
  | var: nat -> LC
  | app: LC -> LC -> LC
  | lam: LC -> LC
.

Then, we

  • formalize the notions of closed terms and beta reduction (e.g. to prove forall s t, (\ \ #1) s t -->* s),
  • define an internalization [[-]] of Coq-level lambda terms into lambda-level terms (Church encoding) and subsequently define an evaluator term f such that f [[x]] -->* x,
  • and in dtlc.v (dependently typed LC) define a type system.

The evaluator is due <insert other person's name here after I received permission to do so>.

Status

In development. Suggestions welcome!