That would be a circular argument: man must cooperate with grace and therefore if a doctrine doesn't teach the ability to cooperate then it's a false doctrine. 

Nowhere does Scripture teach that unregenerate man can or must "cooperate" with the grace of God. Enemies do not cooperate. We need complete rebirth (regeneration).

 The eating of the forbidden fruit was an act of volition which carried with it the threat of death upon the eating. "In the day that you eat of it, dying, you will die." This curse was carried out upon Adam and all his posterity (descending from him by ordinary generation). Man's will was from then forward, being cursed, bent against God. This is what it means that we are "children of wrath by nature," "enemies of God," etc. 

 Man is completely free to choose whatever his nature desires. But the unregenerate man hates God and the things of God (again, see Romans 1 and Ephesians 2 among others).

We do not "cooperate with grace" any more than Lazarus "cooperated" with the call of Christ. All he brought to the table, so to speak, was a carcass.

For a fuller argument, see R. C. Sproul, [Pelagian Captivity of the Church](https://www.monergism.com/pelagian-captivity-church-0).