# Type inference

## Untyped lambda calculus

## Mogensen's encoding

`Efficient Self-Interpretation in Lambda Calculus`, Mogensen

```
-- Predefined:
S  := λn f x.f(n f x)
1  := λf x.f x
Y  := λf.(λx.f(x x)) (λx.f(x x))
R  := λ m . RR m (λa b . b)
E  := Y (λe m.m (λx.x) (λm n.(e m)(e n)) (λm v.e (m v)))
P  := Y (λp m. (λx.x (λv.p (λa b c.b m (v (λa b.b))))m))
RR := Y (λr m.m (λx.x)
         (λm n.
          (r m)
           (λa b.a)
            (r n))
             (λm.(λg x.x g
              (λa b c.c
               (λw.g (P
                (λa b c.a w))
                 (λa b.b))))
                  (λv.r (m v))))

-- Examples:
E (quote (S (S (S 1))))        -- (1) Self-interpreter demo
R (quote (S (S (S 1))))        -- (2) Self-reducer demo

-- Result (1)
λv v1.v (v (v (v v1)))

-- Result (2)
λa b c . c
 (λw a b c . c
  (λw1 a b c . b
   (λa b c . a w)
    (λa b c . b
     (λa b c . a w)
      (λa b c . b
       (λa b c . a w)
        (λa b c . b
         (λa b c . a w)
          (λa b c . a w1))))))
```

### Quote

```
Var: λm . λa . λb . λc . a m
App: λm . λn . λa . λb . λc . b m n
Lam: λm .      λa . λb . λc . c m

-- (1) Our `quote` primitive lies outside lambda calculus:
quote ((λy . y) x)

-- (2) Encoding terms in lambda calculus (without typing the raw encoding):
App (Lam (λy . Var y)) (Var x)

-- output
λ a b c . b (λ a b c . c (λ y a b c . a y)) (λ a b c . a x)   -- (1)
λ a b c . b (λ a b c . c (λ y a b c . a y)) (λ a b c . a x)   -- (2)
```

### Shallow encoding

```
Var: λx . λi . x
App: λm . λn . λi . i (m i) (n i)
Lam: λf . λi . λx . f x i

-- self-interpreter:
E: λq . q (λx . x)

qsucc: Lam (λn .
        Lam (λf .
         Lam (λx .
          App (Var f)
            (App
             (App
              (Var n)
               (Var f))
                (Var x)))))

q0: Lam (λf .
     Lam (λx .
      Var x))

-- Compute `succ (succ 0)`
E(App qsucc (App qsucc q0))
```

### Factorial

```
true = \x y -> x
false = \x y -> y

0 = \f x -> x
1 = \f x -> f x

succ = \n f x -> f(n f x)
pred = \n f x -> n(\g h -> h (g f)) (\u -> x) (\u ->u)

mul = \m n f -> m(n f)
is0 = \n -> n (\x -> false) true

Y = \f -> (\x -> x x)(\x -> f(x x))

fact = Y(\f n -> (is0 n) 1 (mul n (f (pred n))))

-- Computes (4!)
fact (succ (succ (succ 1)))
```


---

# Agent Instructions: 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/lambda-calculus-encodings.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.