

The GCD and the minimum  
It all began with a friend who was preparing 



(All variables are of type natural and ↓ 



Having learned to like Lattice Theory and 



From the above definition, many nice 







As the above are general results from element 

Besides ↓, (0) contains the special con 





and I postponed the choice. 

Note At the time I did not realize that (4) 

(4) connects * with 1 but I assumed 





Now the proof of (0) was straightforward: 



which completes the proof of (0). 



The above proof is nice, but one can object 



where x,y,z are of type real and ↓ denotes 



the minimum. For this proof I used — besides 



which holds because ≤ is a total order 



— i.e. + distributes over ↓ —, which follows 

In order to prove (7) we observe 



We failed to prove (7) because it is not a 



theorem: it only holds for nonnegative x,y,z 



From (*) we first eliminated ↓ at the left 



and 



from which we conclude 





a beautiful formula that I did not know, 

Because, in contrast to ≤, ⊑ in the 

Acknowledgement I thank Hamilton Richards 

Austin, 27 November 2001 



Prof. Dr. Edsger W. Dijkstra 
Transcriber: John S. Adair
Copy Editor: Ham Richards
Last revised on Thu., 22 May 2003.