> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/340-lambda-calculi/lambda-calculus-reductions/eta-conversion3.md).

# Lambda calculus: η-conversion

\-- y is free (λx.xy) -- so let's lift the exp to bind it λy.(λx.xy) ≡ λyx.xy

\-- in haskell f x y = map j \[] y ≡ f x = map j \[]

\-- η-introduction -- x,y are formal params -- A is previously defined final concrete value -- I is id function (λx. x) A ≡ A (λx. A) A ≡ A (λx. A) y ≡ A

`x` ≡ `(λ_. x) _` x ≡ (λv. x) v x ≡ (λv. v) x

λv. (x x) v)

```js
// 1) direct val
'bingo!'
// 2) delayed value x1: thunk 1
(() => 'bingo!')()
// 2) delayed value x2: thunk 1+1
(() => (() => 'bingo!') ()) ()
// 2) delayed value x3: thunk 1+1+1
(() => (() => (() => 'bingo!') ()) ()) ()
// 2) delayed value x4: thunk 1+1+1+1
( () => (
        () => (
            () => (
                    () => 'bingo!'
                  ) ()
              ) ()
          ) ()
) ()


//
(v => (v => 3) (v)) ()

(c => (b => (a => 3) (b)) (c)) ()



//
(c =>
    (b =>
        (a => 'bingo!')
        (b)
    )
    (c)
)
(2)
```


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://mandober.gitbook.io/math-debrief/340-lambda-calculi/lambda-calculus-reductions/eta-conversion3.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.