Their immensely long proof made extensive use of a computer, and involved checking more than 1,000 special cases.Another of Mr Casti's five is even older.
Some comments will be helpful for understanding the long proof of Theorem 2.8.
In the following, we will divide the rather long proof into three steps.
(11) See [4], Theorem 4.1, pp.3-4, for a direct (but long) proof of (11) by using the spectral mapping theorem and some existing lemmas.
For the elementary (but long) proof, we refer to [17, Lemma 2.3].
The mathematician Alan Turing, he noted, had shown long ago that even very short theorems could have very long proofs, running hundreds of pages.
Interestingly, it dates from the same year as the construction, due to Rosser, that eliminates the use of ω-consistency in the first Incompleteness Theorem; like the Speed-up Theorem of Gödel, Rosser's construction exploits the issue of short and long proofs.
But neither option would be politically sustainable for long without proof of genuine progress.
His study provided gene therapy advocates with what they have long lacked: proof, in principle, that the concept can work.
And, though we haven't been forced to pore over the long-form proof of it, we also know that Michelle is forty-seven; Sasha is ten; and Malia, born the same day as our country, is thirteen.
