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تسجيل مستخدم جديدCoaxioms: flexible coinductive definitions by inference systems
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تأليف
Francesco Dagnino
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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 allow us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, whose existence is guaranteed by a smooth extension of classical results. 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, but are rather mixed together.
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After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types. Besides axioms and inference rules with the usual meaning, a generalized inference system allows
coaxioms, which are, intuitively, axioms which can only be applied at infinite depth in a proof tree. This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, while providing more flexibility. Indeed, the classical results can be extended to our generalized framework, interpreting recursive definitions as fixed points which are not necessarily the least, nor the greatest one. This allows formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, or are mixed together.
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This note formally defines the concept of coinductive validity of judgements, and contrasts it with inductive validity. For both notions it shows how a judgement is valid iff it has a formal proof. Finally, it defines and illustrates the notion of a proof by coinduction.
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