A two-level logic perspective on (simultaneous) substitutions

Abstract

Lambda-tree syntax (λTS), also known as higher-order abstract syntax (HOAS), is a representational technique where the pure λ-calculus in a meta-language is used to represent binding constructs in an object language. A key feature of λTS is that capture-avoiding substitution in the object language is represented by β-reduction in the meta language. However, to reason about the meta-theory of (simultaneous) substitutions, it may seem that λTS gets in the way: not only does iterated β-reduction not capture simultaneity, but also β-redexes are not first-class constructs.