What is the ruling on buying a car with a down payment and monthly installments if the remaining amount becomes greater than its cash value in the market, and is it permissible to receive a discount for accelerating the payment of the remaining amount in a single lump sum?

It is not permissible to reduce a debt and waive a part of it in exchange for prompt payment, according to the majority of scholars. This is usury known as "give less and pay sooner." However, if the creditor waives a part of the debt upon its full repayment without prior collusion or established custom, then there is no harm in that.