McCarthy 91 Function - Proof

Proof

Here is a proof that the function behaves as expected.

For 90 ≤ n < 101,

M(n) = M(M(n + 11)) = M(n + 11 - 10), where n + 11 >= 101 since n >= 90 = M(n + 1)

So M(n) = 91 for 90 ≤ n < 101.

We can use this as a base case for induction on blocks of 11 numbers, like so:

Assume that M(n) = 91 for a ≤ n < a + 11.

Then, for any n such that a - 11 ≤ n < a,

M(n) = M(M(n + 11)) = M(91), by hypothesis, since a ≤ n + 11 < a + 11 = 91, by the base case.

Now by induction M(n) = 91 for any n in such a block. There are no holes between the blocks, so M(n) = 91 for n < 101. We can also add n = 101 as a special case.