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:

    “If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
    —Polly Berrien Berends (20th century)

    “The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
    —Charles Baudelaire (1821–1867)

    “The chief contribution of Protestantism to human thought is its massive proof that God is a bore.”
    —H.L. (Henry Lewis)