kill(all);

van=facebook.com/LearningCalculus$

infix("♥")$infix("♦")$

"♥"(a,b):=2*a*b$
"♦"(a,b):=a+b-2$

a ♦ a;
a ♥ a;

expand((a ♥ a) ♦ a);
expand((a ♦ a) ♥ a);

van=facebook.com/LearningCalculus$

      Given a set S and two binary operators ♥ and ♦ on S,
      We say that ♥ is left-distributive over ♦ iff
      given elements x, y and z in S:
      x ♥ (y ♦ z) = (x ♥ y) ♦ (x ♥ z)     

expand(x ♥ (y ♦ z) = (x ♥ y) ♦ (x ♥ z));

is(x ♥ (y ♦ z) = (x ♥ y) ♦ (x ♥ z));

      So this pair of operators is not left-distributive.
      Let's check for right-distributive property:
      ♥ is right-distributive over ♦ iff
      (y ♦ z) ♥ x = (x ♥ y) ♦ (x ♥ z)

expand((y ♦ z) ♥ x = (x ♥ y) ♦ (x ♥ z));

is((y ♦ z) ♥ x = (x ♥ y) ♦ (x ♥ z));

      So this pair of operators is not right-distributive either.
      Neither is either converse:

is(expand(x ♦ (y ♥ z) = (x ♦ y) ♥ (x ♦ z)));
is(expand((y ♥ z) ♦ x = (x ♦ y) ♥ (x ♦ z)));