Safety and conservativity of definitions in HOL and Isabelle/HOL

摘要

De nitions are traditionally considered to be a safe mechanism for introducing concepts on top of a logic known to be consistent. In contrast to arbitrary axioms, de nitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, de nitions should form a conservative extension of the pure logic. These properties are crucial for modern interactive theorem provers, since they ensure the consistency of the logic, as well as a valid environment for total/certi ed functional programming.\ud\udWe prove these properties, namely, safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users. Some unique features of HOL, such as the requirement to give non-emptiness proofs when de ning new types and the impossibility to unfold type de nitions, make the proof of these properties, and also the very formulation of safety, nontrivial.\ud\udOur study also factors in the essential variation of HOL de nitions featured by Isabelle/HOL, a popular member of the HOL-based provers family. The current work improves on recent results which showed a weaker property, consistency of Isabelle/HOL’s de nitions.