The Weak Normalization of the Simply Typed λse-calculus

In this paper, we show the weak normalization (WN) of the simply-typed λse-calculus with open terms where abstractions are decorated with types, and metavariables, de Bruijn indices and updating operators are decorated with environments. We show a proof of WN using the λωe-calculus, a calculus isomorphic to $\lambda\vec{s}_{e}$. This proof is strongly influenced by Goubault-Larrecq's proof of WN for the λ σ-calculus but with subtle differences which show that the two styles require different attention. Furthermore, we give a new calculus λ ω′e which works like λse but which is closer to λ σ than λωe. For both λωe and λ ω′e we prove WN for typed semi-open terms (i.e. those which allow term variables but no substitution variables), unlike the result of Goubault-Larrecq which covered all λ σ open terms.