A Proof for P =? NP Problem

The $\textbf{P}$ vs. $\textbf{NP}$ problem is an important problem in contemporary mathematics and theoretical computer science. Many proofs have been proposed to this problem. This paper proposes a theoretic proof for $\textbf{P}$ vs. $\textbf{NP}$ problem. The central idea of this proof is a recursive definition for Turing machine (shortly TM) that accepts the encoding strings of valid TMs. By the definition, an infinite sequence of TM is constructed, and it is proven that the sequence includes all valid TMs. Based on these TMs, the class $\textbf{D}$ that includes all decidable languages and the union and reduction operators are defined. By constructing a language $\textbf{Up}$ of the union of $\textbf{D}$, it is proved that $\textbf{P}=\textbf{Up}$ and $\textbf{Up}=\textbf{NP}$, and the result $\textbf{P}=\textbf{NP}$ is proven.