<font color="#f00">Why does the cross term appear while folding R1CS instances under the hood? </font>

<br />

To illustrate this issue we split the problem into three parts:

## how does cross term appear

The bottom reason of it is "R1CS constrain system is a relation of vector element-wise **production**, **not addition**", like:

$$
A z \circ B z = C z
$$

If there are two instances to be folded, such as $z_1$ and $z_2$, they have relations:

$$
A z_1 \circ B z_1 = C z_1 \\
A z_2 \circ B z_2 = C z_2 \\
$$

We need a new "bigger" instance $z'$, which contains all information about $z_1$ and $z_2$, also has the same relation:

$$
A z' \circ B z' = C z'
$$

To make it simple(to fold), the new "bigger" instance should be composed through a simple linear combination method:

$$
z' = z_1 + r * z_2
$$

Unfortunately it **DOES NOT** satisfy the original R1CS relation:

$$
A (z_1 + r * z_2) \circ B (z_1 + r * z_2) \overset{?}= C (z_1 + r * z_2)
$$

Therefore the cross term $E$, which have the same number of rows as A/B/C, came out:

$$
A (z_1 + r * z_2) \circ B (z_1 + r * z_2) \overset{?}= C (z_1 + r * z_2) + E
$$

But what's inside the cross term $E$, it's very easy to get:
$$
E = r · (A z_1 ◦ B z_2 + A z_2 ◦ B z_1 − C z_2 − C z_1)
$$

<br />

So we have to update the original R1CS constrain relation into a new **relaxed** one:
$$
A z \circ B z = uC z + E
$$

<br />

## Why does the relaxed relation work

We define two newly instances $U_1 = \{z_1, u_1, E_1\}, U_2 = \{z_2, u_2, E_2\}$, satisfy the updated **relaxed** R1CS relation:

$$
A z_1 \circ B z_1 = u_1 C z_1 + E_1 \\
A z_2 \circ B z_2 = u_2 C z_2 + E_2 \\
$$

Then if we have a new folded "bigger" instance $U' = \{z', u', E'\}$, the first term $z'$:

$$
z' = z_1 + r * z_2
$$

and other two auxilaries:
$$
u' = u_1 + r * u_2 \\
E' = E_1 + r * E_2 \\
$$

So does the newly instance $U'$, which contains $z'$, $u'$ and $E'$, satisfies the **relaxed** R1CS relation? Let's see how it goes:

$$
A z' \circ B z' \overset{?}= u' C z' + E'
$$

The answer is YES, you can try it out:
$$
A (z_1 + r*z_2) \circ B (z_1 + r*z_2) \overset{?}= (u_1 + r * u_2) C (z_1 + r*z_2) + (E_1 + r * E_2)
$$

<br />

## what if it is a addition

If R1CS constrain relation is a addition one:

$$
A z_1 + B z_1 = C z_1 \\
A z_2 + B z_2 = C z_2 \\
$$

and

$$
z' = z_1 + r * z_2 \\
$$

We can certaily have:

$$
A z' + B z' = C z' \\
$$

No cross terms, no **relaxed** relation any more! Unfortunately it's actually not, it won't happen!

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