3 research outputs found
Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading
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.Comment: In Proceedings LFMTP 2020, arXiv:2101.0283
Safety and conservativity of definitions in HOL and Isabelle/HOL
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
Safety and conservativity of definitions in HOL and Isabelle/HOL
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