Theory of complexity – Is there a reduction in polynomial time of a problem NP-difficult to complete (tautology)?

The statement is: If a problem $ P $ is $ mathbf {NP} $-hard, then there is a reduction of $ text {FALSIFIABLE} $ at $ P $. ($ text {FALSIFIABLE} $ being the set of formulas for which there is an assignment that makes the formula false; this is trivially equal to the complement of $ text {TAUT} $.)

This statement is correct.

Why? Simply because $ text {FALSIFIABLE} $ is $ mathbf {NP} $-Achevée. You can prove it exactly the same way $ text {SAT} $ is proven to be $ mathbf {NP} $-complete, by reversing only the truth values ​​"true" and "false" (that is, instead of looking for an assignment that makes the formula true, you are looking for an assignment that makes the false formula).