A Formalised Proof of the Soundness and Completeness of Simply Typed Lambda Calculus with Explicit Substitution

We present a simply-typed λ-calculus with explicit substitutions and we give a fully formalised proof of its soundness and completeness with respect to Kripke models. We further give conversion rules for the calculus and show also for them that they are sound and complete with respect to extensional equality in the Kripke model. A decision algorithm for conversion is given and proven correct. We use the technique "normalisation by evaluation" in order to prove these results, An important aspect of this work is that it is not a formalisation of an existing proof, instead the proof has been done in interaction with the proof system, ALF