# Arithmetization

$s_l^{(1)} x_l^{(1)} + s_l^{(1)}x_l^{(1)} + s_m^{(1)}x_l^{(1)}x_r^{(1)} = s_o^{(1)}x_o^{(1)} + c^{(1)}$
$s_l^{(2)} x_l^{(2)} + s_l^{(2)}x_l^{(2)} + s_m^{(2)}x_l^{(2)}x_r^{(2)} = s_o^{(2)}x_o^{(2)} + c^{(2)}$

.
.
.
$s_l^{(n)} x_l^{(n)} + s_l^{(n)}x_l^{(n)} + s_m^{(n)}x_l^{(n)}x_r^{(n)} = s_o^{(n)}x_o^{(n)} + c^{(n)}$

For the line $i$ we have:

$~s_l^{(i)} ~x_l^{(i)} ~+ ~s_l^{(i)}~x_l^{(i)} ~+ ~s_m^{(i)}~x_l^{(i)}~x_r^{(i)} ~= ~s_o^{(i)}~x_o^{(i)} ~+ ~c^{(i)}$

We define $s_l(\omega^i)=s_l^{(i)}, x_l(\omega^i)=x_l^{(i)}$ etc., so we can rewrite line $i$ above:

$s_l(\omega^i)x_l(\omega^i) + s_l(\omega^i)x_l(\omega^i) + s_m(\omega^i)x_l(\omega^i)x_r(\omega^i) = s_o(\omega^i)x_o(\omega^i) + c(\omega^i)$