Strong Normalization of Substitutions

ασ-calculus is an extended λ-calculus where substitutions are handled explicitly. It is similar to, and inspired by, Categorical Combinatory logic (CCL). The strong normalization of σ, the subcalculus which computes substitutions, may be inferred from the strong normalization of the similar subsystem SUBST of CCL. We present here an independent proof of the termination of several substitution calculi, including σ and SUBST.