We illustrate a methodology for formalizing and reasoning about Abadi and Cardelli's object-based calculi, in (co)inductive type theory, such as the Calculus of (Co)Inductive Constructions, by taking advantage of Natural Deduction Semantics and coinduction in combination with weak Higher-Order Abstract Syntax and the Theory of Contexts. Our methodology allows to implement smoothly the calculi in the target metalanguage; moreover, it suggests novel presentations of the calculi themselves. In detail, we present a compact formalization of the syntax and semantics for the functional and the imperative variants of the ς-calculus. Our approach simplifies the proof of Subject Reduction theorems, which are proved formally in the proof assistant Coq with a relatively small overhead.