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title: Untitled

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For R1CS Constraint with variables $x_1,x_2,\ldots,x_n$, $y$, and $z_1,z_2,\ldots, z_m$,
we denote that the constraint is satisfied iff $CS(x_1,x_2,x_3,\ldots,x_n,y,z_1,\ldots, z_m)=1$

Then, to verify that a function $y = f(x_1,x_2,\ldots,x_n)$ corresponds to the R1CS constraint, we need to verfiy the following two conditions are true:

First,
$$
\forall x_1,x_2,x_3,\ldots, x_n \exists z_{1},z_{2},\ldots,z_{m}\\s.t. Q(x_1,x_2,\ldots,x_n,f(x_1,x_2,\dots, x_n),z_1,z_2,\ldots,z_m)=1
$$
Second,
$$
\forall x_1,x_2,x_3,\ldots,x_n,y,z_1,z_2,\ldots,z_m\\ Q(x_1,x_2,\ldots,x_n,y,z_1,z_2,\ldots,z_m)=1\Rightarrow y=f(x_1,x_2,\ldots,x_n)
$$


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or another formulation:
We want to verify $f(x_1,x_2,\ldots,x_n)=0$ corresponds to constraints $Q(x_1,x_2,x_3,\ldots,x_n,y,z_1,\ldots, z_m)=1$
Then the conditions are

$$
\forall x_1,x_2,x_3,\ldots, x_n \exists z_{1},z_{2},\ldots,z_{m}\\s.t. ,f(x_1,x_2,\dots, x_n)=0 \Rightarrow Q(x_1,x_2,\ldots,x_n,z_1,z_2,\ldots,z_m)=1
$$
and
$$
\forall x_1,x_2,x_3,\ldots,x_n,z_1,z_2,\ldots,z_m\\ Q(x_1,x_2,\ldots,x_n,z_1,z_2,\ldots,z_m)=1\Rightarrow f(x_1,x_2,\ldots,x_n)=0
$$