The P versus NP Problem

Let a decision problem be a function from lists of booleans to booleans. A complexity class is a set of decision problems. A function is computable in polynomial time if there exists a deterministic Turing machine that computes it in time polynomial in the input length. The complexity class P is the set of decision problems computable in polynomial time by a deterministic Turing machine. The complexity class NP is the set of decision problems for which there exists a polynomial $p$ over $\mathbb{N}$ and a polynomial-time computable relation $R : (\text{List } \text{Bool} \times \text{List } \text{Bool}) \to \text{Bool}$ such that for all inputs $x$, the decision problem accepts $x$ if and only if there exists a witness $w$ with $|w| \leq p(|x|)$ such that $R(x, w)$ holds. Prove that $\mathsf{P} \neq \mathsf{NP}$.