What is the ruling on a father's refusal to spend on his children, and is he sinful for that, if the mother files a lawsuit against him for past and future alimony, claiming that he is negligent in spending, even though he is actually spending on them?

It is obligatory for a father to provide for his young children, by consensus, and he is not permitted to withhold maintenance. The Prophet (peace be upon him) warned that whoever neglects those they are responsible to provide for is committing a sin, and that Allah will question every guardian about what they were entrusted with, including a father concerning his household.

A father can prove his provision by obtaining his divorced wife's monthly signed receipt of the maintenance, with witnesses, or by paying the maintenance through the court via what is called an "offer notification" to prove his expenditure. If the father is refusing or failing to provide maintenance, the mother can take the matter to court. However, if he is fulfilling his obligations, it is not permissible to file a lawsuit.