If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time.
Fortune showed in 1979 that if any sparse language is co-NP-complete (or even just co-NP-hard), then P = NP,[1] a critical foundation for Mahaney's theorem.
[2] This means that for every co-NP problem L, there exists a polynomial time algorithm which can transform any instance of L into an instance of C with the same truth value.
One example of a co-NP-complete problem is tautology, the problem of determining whether a given Boolean formula is a tautology; that is, whether every possible assignment of true/false values to variables yields a true statement.
This is closely related to the Boolean satisfiability problem, which asks whether there exists at least one such assignment, and is NP-complete.