In generative syntax, a CP-reduction is a process which deletes CP (also known as S-bar deletion).
In (i) expect assigns structural accusative Case to the subject (Greg) of its complement clause (Greg to win). This should not be possible if this clause were a full CP, because expect could not govern the subject in that case (CP would be a barrier). If expects deletes CP, government is possible, as the remaining IP is not a barrier to government.
(i) Everybody expected [IP Greg to win ]
CP-reduction has also been assumed to obtain in Raising to Subject constructions such as (ii):
(ii) John seems [IP t to have won ]
CP-reduction has been proposed to explain Exceptional Case Marking (ECM).
Chomsky, Noam A. 1981. Lectures on Government and Binding. Dordrecht:Foris.