Adder

template Adder(n) {
  signal input in[n];
  signal output out;

  var lc = 0;
  for (var i = 0; i < n; i++) {
    lc += in[i];
  }
  out <== lc;
}

Adding an array of values and returning the sum is a common operation within circuits.

$$\text{sum}=\sum _{i=0}^{n-1}{a}_i$$

No matter how large $n$ is, a sum operation is just a single linear constraint! Formally, for an array $\vec{a}=[a_0,a_1,\dots ,a_{n-1}]$ with sum $s$ the following is a valid rank-1 constraint:

$$1\cdot (s)=(a_0+a_1+\dots +a_{n-1})$$

The variable lc within the code stands for "linear combination" and its just a variable that stores the expression on the right hand-side in equation above:

lc = in[0] + in[1] + ... + in[n-1]