### eta abstraction ⇝

## eta conversion

<*theory*>

In lambda-calculus, the eta conversion rule states

\ x . f x <--> fprovided x does not occur as a free variable in f and f is a function. Left to right is eta reduction, right to left is eta abstraction (or eta expansion).

This conversion is only valid if bottom and \ x . bottom are equivalent in all contexts. They are certainly equivalent when applied to some argument - they both fail to terminate. If we are allowed to force the evaluation of an expression in any other way, e.g. using seq in Miranda or returning a function as the overall result of a program, then bottom and \ x . bottom will not be equivalent.

See also observational equivalence, reduction.

### Nearby terms:

ET++ ♦ eta abstraction ♦ **eta conversion** ♦ eta expansion ♦ eta reduction

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