Coaxioms: flexible coinductive definitions by inference systems
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We introduce a generalized notion of inference system to support more flexible interpretations of recursive definitions. Besides axioms and inference rules with the usual meaning, we allow also coaxioms, which are, intuitively, axioms which can only be applied "at infinite depth" in a proof tree. Coaxioms allows us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, and classical results, which smoothly extend to this generalized framework, ensure the existence of such fixed points. This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, thus allowing formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, or are mixed together.
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