HOL with Definitions: Semantics, Soundness, and a Verified Implementation
We present a mechanised semantics and soundness proof for the HOL Light kernel including its definitional principles, extending Harrison’s verification of the kernel without definitions. Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system. Our semantics is the first for Wiedijk’s stateless HOL; our implementation, however, is stateful: we give semantics to the stateful inference system by translation to the stateless. We improve on Harrison’s approach by making our model of HOL parametric on the universe of sets. Finally, we prove soundness for an improved principle of constant specification, in the hope of encouraging its adoption. This paper represents the logical kernel aspect of our work on verified HOL implementations; the production of a verified machine-code implementation of the whole system with the kernel as a module will appear separately.
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- 1.Arthan, R.: HOL formalised: Semantics, http://www.lemma-one.com/ProofPower/specs/spc002.pdf
- 2.Arthan, R.: HOL constant definition done right. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS (LNAI), vol. 8558, pp. 531–536. Springer, Heidelberg (2014)Google Scholar
- 3.Barras, B.: Sets in Coq, Coq in sets. J. Formalized Reasoning 3(1) (2010)Google Scholar
- 7.Kumar, R., Myreen, M.O., Norrish, M., Owens, S.: CakeML: a verified implementation of ML. In: Principles of Prog. Lang. (POPL). ACM Press (2014)Google Scholar
- 9.Myreen, M.O., Davis, J.: The reflective Milawa theorem prover is sound (Down to the machine code that runs it). In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS (LNAI), vol. 8558, pp. 421–436. Springer, Heidelberg (2014)Google Scholar
- 10.Myreen, M.O., Owens, S.: Proof-producing translation of higher-order logic into pure and stateful ML. Journal of Functional Programming FirstView (January 2014)Google Scholar
- 12.Norrish, M., Slind, K., et al.: The HOL System: Logic, 3rd edn., http://hol.sourceforge.net/documentation.html
- 15.Wang, Q., Barras, B.: Semantics of intensional type theory extended with decidable equational theories. In: CSL. LIPIcs, vol. 23, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
- 16.Wiedijk, F.: Stateless HOL. In: Types for Proofs and Programs (TYPES). EPTCS, vol. 53 (2009)Google Scholar