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Comments on Arbeitsblatt 5 from o.Prof.Dr.F.L.Bauer.
For the integer function m(x, y) of the integer arguments x and y , given by
|m(x,y) = (x ≥ y → x+1 ▯ x < y → y-1)||(1)|
|(x = y → y+1 ▯ x ≠ y → m(x, m(x-1, y+1))) = m(x, y)||(2).|
Proof. From (1) follows that the first alternative in the left-hand side of (2) may be replaced by x = y → m(x, y) ; the proof is complete when we show that the second alternative can be replaced by x ≠ y → m(x, y) . This letter replacement is allowed as we shall prove
|m(x, m(x-1, y+1)) = m(x, y)||(3)|
|X ≥ y ⇒ (m(x, x) = m(x, y))||(4)|
|m(x-1, y+1) = x and x-1 ≥ y+1 .||(5)|
The above proof has been given, because 33(!) lines of formal text, as dedicated to a proof of this theorem in said Arbeitsblatt 3, is to our tastes a bit unwieldy. We appreciate that the proof in said Arbeitsblatt 3 has been conducted in terms of a limited repertoire of operations that seem suitable for mechanization. When, however, the length of such proofs and their obvious tedium are presented —as is often the case— as conclusive evidence for the necessity of such mechanizations, it should be clear that at least the above example has failed to convince us of such necessity.
|Plataanstraat 5||prof.dr.Edsger W.Dijkstra|
|NL-4565 NUENEN||Burroughs Research Fellow|
Transcribed by Martin P.M. van der Burgt