The GCD and the minimum

It all began with a friend who was preparing
had a nice calculational proof of

 (0) x↓y = 1   ⇒   x↓(y*z)  =  x↓z

(All variables are of type natural and ↓
stands for the greatest common divisor.)
I did not have a nice proof of (0), so I
started to think about it, and then the fun
started. Hence this little note.

 * * *

Having learned to like Lattice Theory and
the Galois Connection, I immediately decided
to regard ↓ as the infimum with respect
to the partial order ⊑, read as "divides".
The greatest common divisor ↓ is then de-
fined by

 (1) u ⊑ x↓y   ≡   u ⊑ x  ∧  u ⊑ y for all x,y,u.

From the above definition, many nice
properties of ↓ elegantly follow, such as

 (2) ↓ is associative, symmetric and idempotent
 (3) x↓y ⊑ x   and   x = x↓y  ≡  x ⊑ y .

As the above are general results from element-
ary lattice theory and have nothing to do
with integer arithmetic, I decided to use them
freely as the need would arise. [It turned
out that, to begin with, that need would
only arise for (2).]

Besides ↓, (0) contains the special con-
stants 1 and *, and the next question
was how to capture their relevant prop-
erties. For 1 I could come up with two re-
lations, connecting 1 to * and to ↓ res-
pectively,

 (4) 1*u = u for all u, and
 (5) 1↓u = 1 for all u,

and I postponed the choice.

Note At the time I did not realize that (4)
is the more promising candidate. By present-
ing 1 as a unit element (of *), it offers a
way of eliminating 1, while (5), which
presents 1 as a zero element (of ↓), does
not offer that service. (End of Note.)

(4) connects * with 1 but I assumed
that this was not enough to capture the
"relevant properties" of *. To connect *
with ↓, I came up with

 (6) x↓(y*z)  =  x↓(x↓y * x↓z) .

Now the proof of (0) was straightforward:
we observe for any x,y,z (satisfying (6)
and (0)'s antecedent)

 x↓(y*z) = { (6) } x↓(x↓y * x↓z) = { antecedent of (0) } x↓(1 * x↓z) = { (4) with u := x↓z } x↓(x↓z) = { ↓ associative and idempotent } x↓z   ,

which completes the proof of (0).

 * * *

The above proof is nice, but one can object
that it does not prove very much, as (6),
which we use, is hardly simpler than the
demonstrandum (0). [It is, for (6) does
not contain 1.] So I started to think
about proving (6), but in order to simpli-
fy matters, I switched to the additive
version

 (7) x↓(y+z)  =  x↓(x↓y + x↓z) ,

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

the minimum. For this proof I used — besides
(1) with ⊑ specialized as ≤ —

 (8) x↓y ≤ u   ≡   x≤u  ∨  y≤u for all x,y,u

which holds because ≤ is a total order
— i.e. x≤y  ∨  y≤x — and

 (9) u + x↓y  =  (u+x)↓(u+y) for all x,y,u

— i.e. + distributes over ↓ —, which follows
from (1).

In order to prove (7) we observe

 x↓(y+z)  =  x↓(x↓y + x↓z) ≡ { + over ↓, i.e. (9) thrice } x↓(y+z)  =  x↓(y+z)↓(x+x)↓(x+y)↓(x+z) ≡ { (3) } x↓(y+z)  ≤  (x+x)↓(x+y)↓(x+z) (*) ≡ { (8) } x  ≤  (x+x)↓(x+y)↓(x+z)  ∨ (y+z)  ≤  (x+x)↓(x+y)↓(x+z) ≡ { (1), 2 times twice } ( x ≤ x+x  ∧  x ≤ x+y  ∧  x ≤ x+z )   ∨ ( y+z ≤ x+x  ∧  y+z ≤ x+y  ∧  y+z ≤ x+z ) ≡ { simplifications } ( 0 ≤ x  ∧  0 ≤ y  ∧  0 ≤ z )   ∨ ( z ≤ x  ∧  y ≤ x )

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

theorem: it only holds for nonnegative x,y,z
or when x is the largest of the three.

 * * *

From (*) we first eliminated ↓ at the left-
hand side, using (8), and than [sic] we used
(1) to eliminate the ↓s at the right. We
could have done it in the other order. We
illustrate the consequences with a simpler
example. Consider the calculations

 x↓y  ≤  a↓b ≡ { (8) } x  ≤  a↓b   ∨   y  ≤  a↓b ≡ { (1) } (x ≤ a  ∧  x ≤ b)   ∨   (y ≤ a  ∧  y ≤ b)

and

 x↓y  ≤  a↓b ≡ { (1) } x↓y  ≤  a   ∧   x↓y  ≤  b ≡ { (8) } (x ≤ a  ∨  y ≤ a)   ∧   (x ≤ b  ∨  y ≤ b) ,

from which we conclude

 (x ≤ a  ∧  x ≤ b)   ∨   (y ≤ a  ∧  y ≤ b)   ≡ (x ≤ a  ∨  y ≤ a)   ∧   (x ≤ b  ∨  y ≤ b)

a beautiful formula that I did not know,
but don't try to prove it by predicate cal-
culus alone, for you need that ≤ is a
total order.

Because, in contrast to ≤, ⊑ in the
meaning of "divides" is not a total order
we can expect (6) to be essentially harder
to prove than its additive version (7). I
would not be amazed if the uniqueness
of the prime factorization were needed.

Acknowledgement I thank Hamilton Richards
for starting me on these investigations and
Jayadev Misra for his interest shown.

Austin, 27 November 2001

Prof. Dr. Edsger W. Dijkstra
Department of Computer Sciences
The University of Texas at Austin
Austin, TX 78712-1188
USA