This paper is a contribution to the meta-theory of systems featuring syntax with bindings, such as λ-calculi and logics. It provides a general criterion that targets inductively defined rule-based systems, enabling for them inductive proofs that leverage Barendregt's variable convention of keeping the bound and free variables disjoint. It improves on the state of the art by (1) achieving high generality in the style of Knaster–Tarski fixed point definitions (as opposed to imposing syntactic formats), (2) capturing systems of interest without modifications, and (3) accommodating infinitary syntax and non-equivariant predicates.
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