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Not a first order example but I've recently been working on formalizations of Intruder Deduction and Observer Theory and related proofs of some basic lemmas on consistency of the theories, which I can soon contribute as an example when the issue #77 gets resolved :) I think I've hit another bug.
Some new examples have been proposed for natural numbers, addition, commutativity, etc. by @lambdacalculator.
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