A: Right:
<pre><code>let Y = \T. (\x. T (x x)) (\x. T (x x)) in
-Y Y ≡ \T. (\x. T (x x)) (\x. T (x x)) Y
+Y Y
+≡ \T. (\x. T (x x)) (\x. T (x x)) Y
~~> (\x. Y (x x)) (\x. Y (x x))
~~> Y ((\x. Y (x x)) (\x. Y (x x)))
~~> Y (Y ((\x. Y (x x)) (\x. Y (x x))))
-~~> Y (Y (Y (...(Y (Y Y))...)))</code></pre>
+~~> Y (Y (Y (...(Y (Y Y))...)))
+</code></pre>
#Q: Ouch! Stop hurting my brain.#
<pre><code>let succ = \n s z. s (n s z) in
let X = (\x. succ (x x)) (\x. succ (x x)) in
succ X
-≡ succ ( (\x. succ (x x)) (\x. succ (x x)) )
+≡ succ ( (\x. succ (x x)) (\x. succ (x x)) )
~~> succ (succ ( (\x. succ (x x)) (\x. succ (x x)) ))
-≡ succ (succ X)
+≡ succ (succ X)
</code></pre>
You should see the close similarity with `Y Y` here.
<pre><code>[same definitions]
succ X
-≡ (\n s z. s (n s z)) X
-~~> \s z. s (X s z)
+≡ (\n s z. s (n s z)) X
+~~> \s z. s (X s z)
<~~> succ (\s z. s (X s z)) ; using fixed-point reasoning
-~~> \s z. s ([succ (\s z. s (X s z))] s z)
-~~> \s z. s ([\s z. s ([succ (\s z. s (X s z))] s z)] s z)
-~~> \s z. s (s (succ (\s z. s (X s z))))
+≡ (\n s z. s (n s z)) (\s z. s (X s z))
+~~> \s z. s ((\s z. s (X s z)) s z)
+~~> \s z. s (s (X s z))
</code></pre>
So `succ X` looks like a numeral: it takes two arguments, `s` and `z`,
<pre><code>let prefact = \f n. iszero n 1 (mul n (f (pred n))) in
let fact = Y prefact in
fact 2
-≡ [(\f. (\x. f (x x)) (\x. f (x x))) prefact] 2
+≡ [(\f. (\x. f (x x)) (\x. f (x x))) prefact] 2
~~> [(\x. prefact (x x)) (\x. prefact (x x))] 2
~~> [prefact ((\x. prefact (x x)) (\x. prefact (x x)))] 2
~~> [prefact (prefact ((\x. prefact (x x)) (\x. prefact (x x))))] 2
-≡ [ (\f n. iszero n 1 (mul n (f (pred n)))) (prefact ((\x. prefact (x x)) (\x. prefact (x x))))] 2
+≡ [ (\f n. iszero n 1 (mul n (f (pred n)))) (prefact ((\x. prefact (x x)) (\x. prefact (x x))))] 2
~~> [\n. iszero n 1 (mul n ([prefact ((\x. prefact (x x)) (\x. prefact (x x)))] (pred n)))] 2
~~> iszero 2 1 (mul 2 ([prefact ((\x. prefact (x x)) (\x. prefact (x x)))] (pred 2)))
~~> mul 2 ([prefact ((\x. prefact (x x)) (\x. prefact (x x)))] 1)
...
~~> mul 2 (mul 1 ([prefact ((\x. prefact (x x)) (\x. prefact (x x)))] 0))
-≡ mul 2 (mul 1 (iszero 0 1 ([prefact ((\x. prefact (x x)) (\x. prefact (x x)))] (pred 0))))
+≡ mul 2 (mul 1 (iszero 0 1 (mul 1 ([prefact ((\x. prefact (x x)) (\x. prefact (x x)))] (pred 0)))))
~~> mul 2 (mul 1 1)
~~> mul 2 1
~~> 2