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A Consistent Foundation for Isabelle/HOL

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Interactive Theorem Proving (ITP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9236))

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The interactive theorem prover Isabelle/HOL is based on the well understood Higher-Order Logic (HOL), which is widely believed to be consistent (and provably consistent in set theory by a standard semantic argument). However, Isabelle/HOL brings its own personal touch to HOL: overloaded constant definitions, used to achieve Haskell-like type classes in the user space. These features are a delight for the users, but unfortunately are not easy to get right as an extension of HOL—they have a history of inconsistent behavior. It has been an open question under which criteria overloaded constant definitions and type definitions can be combined together while still guaranteeing consistency. This paper presents a solution to this problem: non-overlapping definitions and termination of the definition-dependency relation (tracked not only through constants but also through types) ensures relative consistency of Isabelle/HOL.

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  1. 1.

    In Isabelle/HOL, as in any HOL-based prover, the “datatype” command is not primitive, but is compiled into “typedef.”

  2. 2.

    This example works in Isabelle2014; our correction patch [1] based on the results of this paper and in its predecessor [19] is being evaluated at the Isabelle headquarters.

  3. 3.

    Namely, Coq 8.4pl5; the inconsistency is fixed in Coq 8.5 beta.

  4. 4.

    The deduction in polymorphic HOL takes place using open formulas in contexts. In addition, Isabelle/HOL distinguishes between theory contexts and proof contexts. We ignore these aspects in our presentation here, since they do not affect our consistency argument.

  5. 5.

    Any infinite (not necessarily countable) set would do here; we only choose \(\mathbb {N}\) for simplicity.

  6. 6.

    Composability reduces the search space when we are looking for the cycle—it tells us that there exist three cases on how to extend a path (to possibly close a cycle): in two cases we can still (easily) extend the path (\(v \le u'\) or \(u' \le v\)) and in one case we cannot (\(v \, {\#}\, u'\)). The fourth case (v and \(u'\) have a non-trivial common instance; formally \(u' \not \le v\) and \(v \not \le u'\) and there exists w such that \(w\le u'\), \(w\le v\)), which complicates the extension of the path, is ruled out by composability. More about composability can be found in the original paper.

  7. 7.

    The correctness proof is relatively general and works for any

    figure q

    on a set \(\mathcal {U}_\varSigma \) endowed with a certain structure—namely, three functions \(= \,: \mathcal {U}_\varSigma \rightarrow \mathcal {U}_\varSigma \rightarrow \mathsf{bool}\), \(\mathsf {App}: (\mathsf {{Type}}\rightarrow \mathsf {{Type}}) \rightarrow \mathcal {U}_\varSigma \rightarrow \mathcal {U}_\varSigma \) and \({\mathsf {size}}: \mathcal {U}_\varSigma \rightarrow \mathbb {N}\), indicating how to compare for equality, type-substitute and measure the elements of \(\mathcal {U}_\varSigma \). In this paper, we set \(\varSigma = (K,\mathsf {arOf},C,{{\mathsf {tpOf}}})\) and \(\mathcal {U}_\varSigma = {{\mathsf {Type}^{\bullet }}}\cup {{\mathsf {CInst}}}^\bullet \). The definition of \(=, \mathsf {App}\) and \({\mathsf {size}}\) is then straightforward: two elements of \({{\mathsf {Type}^{\bullet }}}\cup {{\mathsf {CInst}}}^\bullet \) are equal iff they are both constant instances and they are equal or they are both types and they are equal; \(\mathsf {App}\,\rho \,\tau = \rho (\tau )\) and \(\mathsf {App}\,\rho \,c_{\tau } = c_{\rho (\tau )}\); finally, \({\mathsf {size}}(\tau )\) counts the number of type constructors in \(\tau \) and \({\mathsf {size}}(c_\tau ) = {\mathsf {size}}(\tau )\).



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We thank Tobias Nipkow, Larry Paulson and Makarius Wenzel for inspiring discussions and the anonymous referees for many useful comments. This paper was partially supported by the DFG project Security Type Systems and Deduction (grant Ni 491/13-3) as part of the program Reliably Secure Software Systems (RS3, priority program 1496).

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Correspondence to Ondřej Kunčar .

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Kunčar, O., Popescu, A. (2015). A Consistent Foundation for Isabelle/HOL. In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham.

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