What reasoning rules can be used for the deduction of bisimulation formulas in coalgebraic specifications is problematic because those rules used in algebraic specifications possibly cannot be applied to bisimulation formulas. Although some categorical bisimulation proof methods for coalgebras have been proposed, they are not based on specification languages of coalgebras so that they cannot be used as reasoning rules. In this paper, a specification language based on paths of polynomial functors is proposed to specify polynomial coalgebras. Paths of polynomial functors give detailed observations and transitions on the state space of coalgebras so that the techniques used in transition system specifications can be applied to such a path-based language. In particular, because bisimulations can be characterized by paths, the notions of progressions, respectful functions and faithful contexts can be defined based on paths, and then bisimulation up-to proof techniques, including bisimulation up-to bisimilarities and up-to contexts for transition systems can be transformed into reasoning rules in the language. Several examples illustrate how to reason syntactically about bisimulations in the language by using the rules induced by the bisimulation proof techniques.