HOL-λσ: An Intentional First-Order Expression of Higher-Order Logic

We propose a first-order presentation of higher-order logic based on explicit substitutions. It is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, i.e. a proposition can be proved without the extensionality axioms in one theory if and only if it can in the other. The tiExtended Narrowing and Resolution first-order proof-search method can be applied to this theory. This allows to simulate higher-order resolution step by step and furthermore leaves room for further optimizations and extensions.