Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

TL;DR

This paper presents a mechanized approach to ensure that extending higher-order logic with ad-hoc overloading preserves existing interpretations, maintaining consistency and generalizing previous model constructions.

Contribution

It introduces a mechanized method for model-theoretic conservative extension in HOL with ad-hoc overloading, extending previous theories and ensuring consistency.

Findings

Symbols independent of new definitions retain their interpretation in model extensions

The approach generalizes earlier model constructions for HOL

Consistency of theories with ad-hoc overloading is established

Abstract

Definitions of new symbols merely abbreviate expressions in logical frameworks, and no new facts (regarding previously defined symbols) should hold because of a new definition. In Isabelle/HOL, definable symbols are types and constants. The latter may be ad-hoc overloaded, i.e. have different definitions for non-overlapping types. We prove that symbols that are independent of a new definition may keep their interpretation in a model extension. This work revises our earlier notion of model-theoretic conservative extension and generalises an earlier model construction. We obtain consistency of theories of definitions in higher-order logic (HOL) with ad-hoc overloading as a corollary. Our results are mechanised in the HOL4 theorem prover.