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.

中文翻译：

---

P = 的证明 NP问题

$\textbf{P}$ vs. $\textbf{NP}$ 问题是当代数学和理论计算机科学中的一个重要问题。针对这个问题已经提出了许多证明。本文提出了$\textbf{P}$ vs. $\textbf{NP}$ 问题的理论证明。这个证明的中心思想是图灵机（简称 TM）的递归定义，它接受有效 TM 的编码字符串。根据定义，构造了一个无限的 TM 序列，证明该序列包含所有有效的 TM。基于这些 TM，定义了包含所有可判定语言以及联合和归约运算符的类 $\textbf{D}$。通过构造$\textbf{D}$并集的语言$\textbf{Up}$，证明$\textbf{P}=\textbf{Up}$和$\textbf{Up}=\textbf {NP}$，结果 $\textbf{P}=\textbf{NP}$ 得证。