The function defined by

```
val f = n => if n>100 then n-10 else
f(f(n+11))
```

has value max(`n`-10,91).
We prove this by strong induction on the quantity 101-`n`.

The base case is 101-`n` ≤ 0. Here, clearly, the
claim holds as the function returns `n`-10 which is
greater than or equal to 91.

Now, inductively, suppose that the claim holds for all
101-`n` less than 101-`k`, that is, for
all `n` > `k`.

If 0 ≤ 101-`k` ≤ 11, we have that f(`k`) =
f(f(`k`+11)) = f(`k`+11-10) =
f(`k`+1) which *by our
induction hypothesis* is 91.

If on the other hand 11 < 101-`k`, then f(`k`) =
f(f(`k`+11)) = f(91) *by our
induction hypothesis* = 91.