(1) The subject "a program written in 1956" is vague. What program? Does it have a name? Who wrote it? Later you write "one of them" when referring to the proof. Which one?

(2) The next three sentences start the same way. Try to avoid that.

(3) Omit needless words.

(4) Avoid the passive voice.

(5) This section needs a rewrite.

(6) The adverb is not your friend.

(7) What is a "technological singularity"? You haven't defined this term.

Great start. I look forward to more.