A complete language for $\mathbf{EXPTIME}$

Definition 14   $L = \{(N, 1^{\vert I\vert^k}, I)\mid N\textrm{ is an exp time machine that accepts }I\textrm{ in }2^{\vert I\vert^k}\textrm{ steps}\}$

Theorem 11   $L$ is complete for $\mathbf{EXPTIME}$.

Proof. $\forall x \in \mathbf{EXPTIME}, x \leq L$. $\exists$ Turing Machine $M$ and a $k$ such that $M$ decides $x$ in time $2^{n^k}$.

Polynomial $x$: read $I$, Decide if $M$ accepts $I$. Call program for $L$ on input $(M, 1^{\vert I\vert^k}, I)$. $\qedsymbol$