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.

Read more about this topic:  McCarthy 91 Function

Famous quotes containing the word proof:

    “Right and proof are two crutches for everything bent and crooked that limps along.”
    —Franz Grillparzer (1791–1872)

    “Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
    —M.F.K. Fisher (1908–1992)