<theory> In lambda-calculus extended with constants, delta reduction replaces a function applied to the required number of arguments (a redex) by a result.

E.g. plus 2 3 --> 5. In contrast with beta reduction (the only kind of reduction in the pure lambda-calculus) the result is not formed simply by textual substitution of arguments into the body of a function.

Instead, a delta redex is matched against the left hand side of all delta rules and is replaced by the right hand side of the (first) matching rule.

There is notionally one delta rule for each possible combination of function and arguments.

Where this implies an infinite number of rules, the result is usually defined by reference to some external system such as mathematical addition or the hardware operations of some computer.

For other types, all rules can be given explicitly, for example Boolean negation: