Given four real numbers $x_1,x_2,y_1,y_2\in\mathbb R$ I wonder if in general:
$$ (x_1\lor y_1) (x_2 \lor y_2) + (x_1 \land y_1) (x_2\land y_2) \geq x_1 x_2+y_1 y_2\,.$$
When we restrict ourself to binary variables $x_1,x_2,y_1,y_2\in\{-1,1\}$ the previous inequality can be checked directly (any smarter idea?). Is it still true for real variables?