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There is linear $\mu$-calculus (see e.g. [1]) and first-order $\mu$-calculus (see e.g. here).

Has anybody studied first-order linear $\mu$-calculus?

[1]: Christian Dax, Martin Hofmann, Martin Lange: A Proof System for the Linear Time µ-Calculus. FSTTCS 2006: 273-284

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