etaexpansion

Eta-expansion is a transformation in lambda calculus and functional programming that converts a function value into a new function that takes an argument and applies the original function to that argument. Concretely, if f has type A -> B, eta-expansion yields the function λx:A. f x, so that f and λx. f x are extensionally equivalent under the usual conditions (provided x is not free in f). The opposite transformation is eta-reduction, which removes a lambda when the body is an application of the bound variable to a function.

In languages with currying, eta-expansion makes the parameter structure explicit or adapts functions to higher-order interfaces. For instance, in Haskell a function f :: a -> b is considered η-equivalent to the lambda expression \x -> f x.

In practice, eta-expansion can influence compiler optimizations and runtime performance. In lazy (non-strict) languages, introducing an extra lambda can create additional thunks and hinder inlining, while in strict languages it can cause extra allocations. Therefore compilers apply eta-expansion or its opposite with consideration for performance characteristics and type expectations.

Use cases include adapting a function to a more explicit arity, satisfying particular type signatures or higher-order function expectations, and reasoning about function extensionality. While η-expansion preserves semantics under common equational theories, it remains subject to practical considerations of evaluation strategy and implementation.