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In first-order logic, the formula-based instances of the Noetherian induction principle allow to perform effectively simultaneous, mutual and lazy induction reasoning. Compared to the term-based Noetherian induction instances, they are not directly supported by the current proof assistants.

We provide general formal tools for certifying formula-based Noetherian induction proofs by the Coq proof assistant, then show how to apply them to certify proofs of conjectures about conditional specifications, built with: i) a reductive rewrite-based induction system, and ii) a reductive-free cyclic induction system. The generation of reductive proofs and their certification process can be easily automatised, without requiring additional definitions or proof transformations, but may involve many ordering constraints to be checked during the certification process. On the other hand, the reductive-free proofs generate fewer ordering …