-- Define the ring of Boolean functions on 5 nodes

Q = ZZ/2[x1,x2,x3,x4,x5] /  ideal(x1^2-x1,x2^2-x2,x3^2-x3,x4^2-x4,x5^2-x5); 

-- Shortcut for AND and OR functions

RingElement | RingElement :=(x,y)->x+y+x*y;
RingElement & RingElement :=(x,y)->x*y;

-- Define the Boolean functions

f1 = x5 | (1+x2) | x4;
f2 = (1+x1) & (1+x3);
f3 = 1+x2;
f4 = 1+x2;
f5 = x1 | x4;

-- Eliminate x5. It only appears in f1.

f1=sub(f1,{x5=>f5});

-- Eliminate x4. It only appears in f1.

f1=sub(f1,{x4=>f4});

-- Eliminate x3, which appears in f1 and f2.

f1=sub(f1,{x3=>f3});
f2=sub(f2,{x3=>f3});

-- We can't eliminate further; f1 contains x1 and f2 contains x2.

(f1, f2)