gcd is:exact

gcd x y is the non-negative factor of both x and y of which every common factor of x and y is also a factor; for example gcd 4 2 = 2, gcd (-4) 6 = 2, gcd 0 4 = 4. gcd 0 0 = 0. (That is, the common divisor that is "greatest" in the divisibility preordering.) Note: Since for signed fixed-width integer types, abs minBound < 0, the result may be negative if one of the arguments is minBound (and necessarily is if the other is 0 or minBound) for such types.

The Greatest Common Divisor is defined by:

gcd x y == gcd y x
divides z x && divides z y ==> divides z (gcd x y)   (specification)
divides (gcd x y) x

Greatest common divisor. Must satisfy

\x y -> isJust (x `divide` gcd x y) && isJust (y `divide` gcd x y)

\x y z -> isJust (gcd (x * z) (y * z) `divide` z)

Greatest common divisor. Must satisfy

\x y -> isJust (x `divide` gcd x y) && isJust (y `divide` gcd x y)

\x y z -> isJust (gcd (x * z) (y * z) `divide` z)

gcd x y is the non-negative factor of both x and y of which every common factor of x and y is also a factor; for example gcd 4 2 = 2, gcd (-4) 6 = 2, gcd 0 4 = 4. gcd 0 0 = 0. (That is, the common divisor that is "greatest" in the divisibility preordering.) Note: Since for signed fixed-width integer types, abs minBound < 0, the result may be negative if one of the arguments is minBound (and necessarily is if the other is 0 or minBound) for such types.

>>> gcd 72 60
12

Symbolic GCD as our specification. Note that we cannot really implement the GCD function since it is not symbolically terminating. So, we instead uninterpret and axiomatize it below. NB. The concrete part of the definition is only used in calls to traceExecution and is not needed for the proof. If you don't need to call traceExecution, you can simply ignore that part and directly uninterpret. In that case, we simply use Prelude's version.