We present FJ&$lambda$, a new core calculus that extends Featherweight Java (FJ) with interfaces, supporting multiple inheritance in a restricted form, $lambda$-expressions, and intersection types. Our main goal is to formalise how lambdas and intersection types are grafted on Java 8, by studying their properties in a formal setting. We show how intersection types play a significant role in several cases, in particular in the typecast of a $lambda$-expression and in the typing of conditional expressions. We also embody interface emph{default methods} in FJ&$lambda$, since they increase the dynamism of $lambda$-expressions, by allowing these methods to be called on $lambda$-expressions. The crucial point in Java 8 and in our calculus is that $lambda$-expressions can have various types according to the context requirements (target types): indeed, Java code does not compile when $lambda$-expressions come without target types. In particular, in the operational semantics we must record target types by decorating $lambda$-expressions, otherwise they would be lost in the runtime expressions. We prove the subject reduction property and progress for the resulting calculus, and we give a type inference algorithm that returns the type of a given program if it is well typed. The design of FJ&$lambda$ has been driven by the aim of making it a subset of Java 8, while preserving the elegance and compactness of FJ. Indeed, FJ&$lambda$ programs are typed and behave the same as Java programs.
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