Abstract
Definitions 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, definitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, definitions 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/certified functional programming. We 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 defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also the very formulation of safety, nontrivial. Our study also factors in the essential variation of HOL definitions 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 definitions.
Highlights
Higher-Order Logic (HOL) [Pitts 1993] is an important logic in the theorem proving community
Safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users
1 INTRODUCTION Higher-Order Logic (HOL) [Pitts 1993] is an important logic in the theorem proving community. It forms the basis of several interactive theorem provers, including HOL4 [Gordon and Melham 1993; Slind and Norrish 2008], HOL Light [Harrison 1996], Isabelle/HOL [Nipkow and Klein 2014; Nipkow et al 2002], ProofPower-HOL [Arthan 2004] and HOL Zero [Adams 2010]
Summary
Higher-Order Logic (HOL) [Pitts 1993] (recalled in Section 3 of this paper) is an important logic in the theorem proving community. In HOL-based provers, programming is supported by a different mechanism: All recursive datatype specifications and all recursive specifications of functions on these datatypes are translated into nonrecursive HOL primitives, i.e., constant and type definitions; the recursive specifications are proved automatically as theorems in the logic This scheme involves a massive background compilation and proof process (supported by tools consisting of tens of thousands of lines of code, e.g., [Blanchette et al 2014; Krauss 2009; Melham 1989]). For a theory extension Θ1 ⊆ Θ2, he considers (proof-theoretic) conservativity, a property much stronger than preservation of consistency, to be a minimum requirement for deeming a theory extension truly definitional [Wenzel 1997, p.7] He argues for an even stronger notion, meta-safety. The documented implementation is available from [Kunčar and Popescu 2017c]
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