hoops.out = y & y >= z -> x >= z ]]> = y & y >= x -> x = y ]]> = z <-> y >= x ==> z ]]> = y -> y ==> z >= x ==> z ]]> = y -> z ==> x >= z ==> y ]]> = x ]]> = y) ]]> = z) ]]> = z ]]> = y) ]]> = x) ]]> = z) ]]> = x ==> z ]]> = z ]]> = x ==> z) ]]> = 0 ]]> = y) ]]> z >= x ==> z ]]> = y) ]]> x >= z ==> y ]]> y) = y + (y ==> x) ]]> y) ]]> (x + y) ]]> x) ==> x ]]> = y ==> (y + x) ]]> = u) ]]> = (x + y) ==> u ]]> = y) ]]> = x ==> y ]]> y) >= y ]]> = y ==> 0 ]]> (y + z)) = y + (z + ((y + z) ==> x)) ]]> x) ]]> (y + x) ]]> 0 = 0 ]]> = x ==> x ]]> = m(y,x) ]]> x = 0 ]]> (y ==> y) = 0 ]]> (y + (y ==> z)) >= x ==> z ]]> z)) >= z ]]> y) + x >= y ]]> = y ]]> z) >= z ]]> = (x ==> y) ==> y ]]> x >= x ]]> y >= (y v x) ==> y ]]> = y v x ]]> z >= y ==> z ]]> = z v y ]]> = c(x,y) ==> (z v y) ]]> (y + (z + x)) ]]> x = y ==> x ]]> (z v y) = 0 ]]> (x + (x ==> y)) = x ==> y ]]> = u) ]]> = y ==> u ]]> z)) = z + (x + (z ==> y)) ]]> (y + (x + z)) ]]> c(x,y) = x ==> y ]]> y) = x ==> y ]]> = u) ]]> z >= (x + y) ==> u ]]> z) >= y ==> z ]]> z >= x ==> (y ==> z) ]]> y,z + x) = (x ==> y) ==> (y + (z + (y ==> x))) ]]> z)) = x ==> (z + (y + (z ==> x))) ]]> (y ==> z) >= (y + x) ==> z ]]> (y ==> z) >= (x + y) ==> z ]]> z = x ==> (y ==> z) ]]> z = y ==> (x ==> z) ]]> z = x ==> ((x ==> y) ==> z) ]]> (z ==> x)) ==> x ]]> (y ==> (x + y)) = 0 ]]> (y ==> (z + (x + y))) ]]> x = 0 ]]> (y ==> z) = y ==> (x ==> z) ]]> x ]]> (y ==> x) = 0 ]]> (z ==> y)) = x ]]> y) + ((x ==> y) ==> y) = y ]]> = y ==> x ]]> x = x ]]> y = 0 ]]> y) + ((x ==> y) ==> z) = z + (x ==> (z ==> y)) ]]> z) = x + (y ==> (x ==> z)) ]]> y) v z = x ==> ((z ==> (x ==> y)) ==> y) ]]> (y ==> c(x,z)) = y ==> (x ==> z) ]]> y) + (y v x) = y ]]> x) = x ]]> x)) = z + x ]]> (z ==> x)) ==> x ]]> y,x ==> z) = (x ==> y) + (c(x,y) ==> z) ]]> c(x,z) = (x ==> y) ==> (x ==> z) ]]> m(y,z + x) ]]> m(y,x + z) ]]> m(y,z + x) ]]> x) = (x v y) ==> m(z,x) ]]> m(x,y + z) ]]> y) v z = x ==> (y v (z + x)) ]]> y) v z = x ==> (y v (x + z)) ]]> y) v (y v z) = x ==> (y v (x + z)) ]]> c(x,z) = x ==> ((x ==> y) ==> z) ]]> (m(x + y,z) + x) = m(x + y,z) ]]> y) v (y v z) = (x ==> y) v z ]]> x) ==> x) v z ]]> c(x,z) = c(x,y) ==> z ]]> ((x v y) + (m(x + z,u) + (y ==> x))) = m(x + z,u) ]]> (x v y) = (x v z) ==> (x v y) ]]> (y + z))) ==> y = 0 ]]> (x + y)) = x + (y + 0) ]]> (y + z))) ==> z = 0 ]]> ((x ==> (y + z)) ==> y) = 0 ]]> ((x ==> (y + z)) ==> z) = 0 ]]> ((y ==> (z + u)) ==> z)) = x ]]> (m(x,y) ==> y) = 0 ]]> (x + y)) = x + y ]]> (x + y)) + 0) = x + y ]]> (y + (x + z)) ]]> (m(x,y) ==> z) = (x + y) ==> z ]]> (z + u)) ==> (y ==> z)) = x ]]> (x + y)) + (x + 0) = x + y ]]> (m(x,y) ==> z) = x ==> (y ==> z) ]]> (x + y)) + (x + (z ==> (m(z,u) ==> u))) = x + y ]]> m(m(x,y),z) = x ==> (y ==> (m(x,y) + z)) ]]> m(x,y + z) ]]> (x + (y + z)) ]]> y),z) ]]> = m(c(x,z),y) ]]> = m(c(z,x),y) ]]> = m(c(x,y),z) ]]> y,z + (y v x)) = (x ==> y) ==> (y + (z + (y ==> (x ==> y)))) ]]> y)) = x ==> (y + (z + (y ==> x))) ]]> y) + (x ==> ((x ==> y) ==> (y + z))) = x ==> (y + z) ]]> c(y,z),x ==> y) = (x ==> c(y,z)) + 0 ]]> (x v z) = y ==> (x v z) ]]> (y v x) = 0 ]]> (z v y)) = x ]]> (y ==> (z v x)) = y ==> 0 ]]> z = c(y,x) ==> c(x,z) ]]> (y ==> (z v x)) = 0 ]]> (y ==> ((x ==> z) ==> z)) = 0 ]]> c(y,z) = c(x,y) ==> z ]]> y,x ==> c(y,z)) = (x ==> y) + (c(x,y) ==> z) ]]> c(y,z),x ==> y) = x ==> c(y,z) ]]> y,x ==> c(y,z)) = x ==> c(y,z) ]]> (m(z,u) ==> u))) = x + y ]]> z) + ((m(x,z) ==> z) ==> x)) = x + y ]]> y)) = m(x,z + (x ==> y)) ]]> (y v x))) ]]> y,x ==> c(y,z)) = c(x ==> y,x ==> z) ]]> y,x ==> z) = x ==> c(y,z) ]]> c(y,z) = (x ==> z) + ((x ==> z) ==> (x ==> y)) ]]> y) + ((x ==> y) ==> (x ==> z)) = x ==> (y + (y ==> z)) ]]> y) + (x ==> ((x ==> y) ==> z)) = x ==> (y + (y ==> z)) ]]> y) + (x ==> m(x ==> y,z)) = x ==> (y + (y ==> ((x ==> y) + z))) ]]> ((x v y) + ((y ==> x) + m(x + z,u))) = m(x + z,u) ]]> z)) + ((m(x,z) ==> z) ==> x) = x + y ]]> y),z) = x ==> ((x v u) + ((u ==> x) + m(y + (y ==> x),z))) ]]> m(y,x))) + ((m(y,x) ==> x) ==> y) = y + x ]]> (y ==> (x + y)))) + ((m(y,x) ==> x) ==> y) = y + x ]]> x) ==> y) = y + x ]]> x) ==> y) = y + x ]]> (y + x)) ==> x) ==> y) = y + x ]]> y)) + (((z ==> (z + (y + (y ==> x)))) ==> (y + (y ==> x))) ==> z) = z + (y + (y ==> x)) ]]> y,z + (y v x)) = (x ==> y) ==> (y + z) ]]> y,(z + (y ==> x)) + (y v x)) = m(x ==> y,z + x) ]]> y) + (x ==> m(x ==> y,z + (y v x))) = x ==> (y + z) ]]> y) + (x ==> m(x ==> y,(z + (y ==> x)) + (y v x))) = m(x,z + (x ==> y)) ]]> y) + (x ==> m(x ==> y,z + x)) = m(x,z + (x ==> y)) ]]> y) + m(x + (x ==> y),z) = m(x,z + (x ==> y)) ]]> y),z) = x ==> ((x v y) + m(y,z + (y ==> x))) ]]> y),z) = y ==> ((y v x) + m(x,z + (x ==> y))) ]]> x))) + (((x v y) + m(y,z + (y ==> x))) ==> x) = x + m(y + (y ==> x),z) ]]> y) + (((z ==> (z + (y + (y ==> x)))) ==> (y + (y ==> x))) ==> z)) = z + (y + (y ==> x)) ]]> y) + (((x ==> y) ==> y) + (((z ==> (z + y)) ==> (y + (y ==> (x ==> y)))) ==> z)) = z + (y + (y ==> (x ==> y))) ]]> x))) + ((x v y) ==> (m(y,z + (y ==> x)) ==> x)) = x + m(y + (y ==> x),z) ]]> y) + (((x ==> y) ==> y) + (((z ==> (z + y)) ==> y) ==> z)) = z + (y + (y ==> (x ==> y))) ]]> y) + (((x ==> y) ==> y) + (((z ==> (z + y)) ==> y) ==> z)) = z + y ]]> (x + y)) ==> y) ==> y) + (x + (x ==> ((x ==> (x + y)) ==> y))) = x + y ]]> (x + y)) ==> y) ==> y) + (x + 0) = x + y ]]> (x + y)) ==> y) ==> y) + x = x + y ]]> (x + y)) ==> y) ==> y) = x + y ]]> (x + y))) = x + y ]]> (x + y) ]]> = m(y,x) ]]> m(z,x) = y ==> (x v m(z,x)) ]]> x) = y ==> (x v m(z,x)) ]]> x) = y ==> m(z,x) ]]> x) = y ==> (z ==> (m(y,z) + x)) ]]> y) + m((z v x) + (x ==> y),x ==> z) = x ==> (y + (y ==> ((x ==> y) + z))) ]]> (x + y)) + m((z v x) + (x ==> (x + y)),x ==> z) = x ==> ((x + y) + ((x + y) ==> (m(x,y) + z))) ]]> x)) + ((x v y) ==> (m(y,z + (y ==> x)) ==> x))) = x + m(y + (y ==> x),z) ]]> (x + y)) + m((z v x) + (x ==> (x + y)),x ==> z) = x ==> ((x + y) + (x ==> (y ==> (m(x,y) + z)))) ]]> x)) + (m(y,z + (y ==> x)) ==> ((x v y) ==> x))) = x + m(y + (y ==> x),z) ]]> ((x + y) + m((z v x) + m(x,y),x ==> z)) = (x ==> (x + y)) + m((z v x) + (x ==> (x + y)),x ==> z) ]]> x)) + (m(y,z + (y ==> x)) ==> (y ==> x))) = x + m(y + (y ==> x),z) ]]> x)) + (y ==> (m(y,z + (y ==> x)) ==> x))) = x + m(y + (y ==> x),z) ]]> x)) + (y ==> ((z + (y ==> x)) ==> x))) = x + m(y + (y ==> x),z) ]]> x)) + (y ==> (z ==> ((y ==> x) ==> x)))) = x + m(y + (y ==> x),z) ]]> x)) + 0) = x + m(y + (y ==> x),z) ]]> x)) = x + m(y + (y ==> x),z) ]]> (x + (y + m((z v x) + m(x,y),x ==> z))) = (x ==> (x + y)) + m((z v x) + (x ==> (x + y)),x ==> z) ]]> z)) = (x ==> (x + y)) + m((z v x) + (x ==> (x + y)),x ==> z) ]]> z)) = m(x,y) + m((z v x) + (x ==> (x + y)),x ==> z) ]]> z)) = m(x,y) + m((z v x) + m(x,y),x ==> z) ]]> z) = m(x,y + (x ==> z)) ]]> x)) = (x v y) + (m(y,z) + (y ==> x)) ]]> x)) = x + m(y + (y ==> x),z) ]]> x),z) = m(y,z) + x ]]> x),z)) = x ==> (m(y,z) + x) ]]> x),z)) = m(x,m(y,z)) ]]> y),z) >= m(y,m(x,z)) ]]> = m(x,m(y,z)) ]]>