Integrating static analysis and general-purpose theorem proving for termination analysis
Proceedings of the 28th international conference on Software engineering
Efficient execution in an automated reasoning environment
Journal of Functional Programming
Proceedings of the 46th Annual Design Automation Conference
Inference rules for proving the equivalence of recursive procedures
Time for verification
Termination analysis with calling context graphs
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
Refinement and theorem proving
SFM'06 Proceedings of the 6th international conference on Formal Methods for the Design of Computer, Communication, and Software Systems
Ordinals in HOL: transfinite arithmetic up to (and beyond) ω1
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
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Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite that were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8.