**Authors:** A. A. Frempong

By applying differential and integral calculus, this paper covers the principles and procedures for producing the solution of a problem, given the procedure for checking the correctness of the solution of a problem, and vice versa. If one is able to check quickly and completely, the correctness of the solution of a problem, one should also be able to produce the solution of the problem by reversing the order of the steps of the checking process, while using opposite operations in each step. The above principles were applied to four examples from calculus as well as to an example from geometry. Even though in calculus, one normally uses differentiation to check the correctness of an integration result, one will differentiate a function first, and then integrate the derivative to obtain the original function. One will differentiate the trigonometric functions, tan x, cot x, sec x and csc x; followed by integrating each derivative to obtain each original function. The results show that the solution process and the checking process are inverses of each other. In checking the correctness of the solution of a problem, one should produce the complete checking procedure which includes the beginning, the middle, and the end of the problem. Checking only the correctness of the final answer or statement is incomplete checking. To facilitate complete checking, the question should always be posed such that one is compelled to show a complete checking procedure from which the solution procedure can be produced. A general application of P = NP is that, if the correctness of the solution of a problem can be checked quickly and it is difficult to write a solution procedure, then first, one can write a complete checking procedure and reverse the order of the steps of the checking procedure while using opposite operations in each step, to obtain the solution procedure for the problem. Therefore, P is equal to NP. Furthermore, every problem with a complete procedure for checking the correctness of the solution of the problem can be solved in polynomial time.

**Comments:** 11 Pages. Copyright © by A. A. Frempong

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[v1] 2019-08-24 01:19:30

[v2] 2019-08-25 00:49:40

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[v4] 2019-08-31 02:58:43

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