Proof-theoretic Conservativity for HOL with Ad-hoc Overloading

Logical frameworks are often equipped with an extensional mechanism to define new symbols. The definitional mechanism is expected to be conservative, i.e. it shall not introduce new theorems of the original language. The theorem proving framework Isabelle implements a variant of higher-order logic where constants may be ad-hoc overloaded, allowing a constant to have different definitions for non-overlapping types. In this paper we prove soundness and completeness for the logic of Isabelle/HOL with general (Henkin-style) semantics, and we prove model-theoretic and proof-theoretic conservativity for theories of definitions.