Until now, we have only considered constraints that apply over values in a single row. But what if we want to express constraints over multiple rows? For example, we may want to ensure that the difference between the values in two adjacent rows is always the same. Turns out we can implement this as an AIR constraint, as long as the same constraints are applied to all rows. In this section, we will see how to implement this. We will build upon the example in the previous section, where we created a two columns and proved that they are permutations of each other by asserting that the second column looks up all values in the first column exactly once. Here, we will create two columns and prove that not only are they permutations of each other, but also that the second row is a sorted version of the first row. More specifically, the sorted column will contain in order the values [0,num_rows)[0,num\_rows), which means that the difference between every current row and the previous row should be 11. We will go through three iterations, fixing an issue in each iteration.

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First Try

fn evaluate<E: EvalAtRow>(&self, mut eval: E) -> E {
    let unsorted_col = eval.next_trace_mask();
    let [sorted_col_prev_row, sorted_col_curr_row] =
        eval.next_interaction_mask(ORIGINAL_TRACE_IDX, [-1, 0]);

    // New constraint
    eval.add_constraint(
        E::F::one() - (sorted_col_curr_row.clone() - sorted_col_prev_row.clone()),
    );

    eval.add_to_relation(RelationEntry::new(
        &self.lookup_elements,
        E::EF::one(),
        &[unsorted_col],
    ));

    eval.add_to_relation(RelationEntry::new(
        &self.lookup_elements,
        -E::EF::one(),
        &[sorted_col_curr_row],
    ));

    eval.finalize_logup_in_pairs();

    eval
}
Basically, the same logic for creating the trace and LogUp columns are equal to the previous section, so we omit them for brevity. What does change is the evaluate function, where we call next_interaction_mask on the EvalAtRow instance. This function can retrieve values from arbitrary row offsets, which means we can access the previous row value using -1. Since we call this function with offsets [-1, 0], we will retrieve the values for the previous and current rows. Once we have these values, we can now assert that the difference between the current and previous row is always 1 with the constraint: E::F::one() - (sorted_col_curr_row.clone() - sorted_col_prev_row.clone()). But this will fail with a ConstraintsNotSatisfied error, can you see why? (You can try running it yourself here)

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Second Try

The issue was that when calling evaluate on the first row of our trace, the previous row value wraps around to the last row because there are no negative indices. This means that in our example, we are expecting the 0 - 15 = 1 constraint to hold, which is clearly not true. To fix this, we can use the IsFirstColumn preprocessed column that we created in the Preprocessed Trace section. So we will copy over the same code for creating the preprocessed column and modify our new constraint as follows: 
eval.add_constraint(
    (E::F::one() - is_first_col.clone())
        * (E::F::one() - (sorted_col_curr_row.clone() - sorted_col_prev_row.clone())),
);
Now, we have a constraint that is disabled for the first row, which is exactly what we want. Still, however, this will fail with the same ConstraintsNotSatisfied error. (You can run it here)

​
Third Try

So when we were creating CircleEvaluation instances from our BaseColumn instances, the order of the elements that we were creating it with was actually not the order that S-two understands it to be. Instead, it assumes that the values are in the bit-reversed, circle domain order. It’s not important to understand what this order is, specifically, but this does mean that when S-two tries to find the -1 offset when calling evaluate, it will find the previous value assuming that it’s in a different order. This means that when we create a CircleEvaluation instance, we need to convert it to a bit-reversed circle domain order. Thus, every time we create a CircleEvaluation instance, we need to convert the order of the values in the BaseColumn beforehand. 
impl IsFirstColumn {
    pub fn new(log_size: u32) -> Self {
        Self { log_size }
    }

    pub fn gen_column(&self) -> CircleEvaluation<SimdBackend, M31, BitReversedOrder> {
        let mut col = BaseColumn::zeros(1 << self.log_size);
        col.set(0, M31::from(1));

        //////////////////////////////////////////////////////////////
        // Convert the columns to bit-reversed circle domain order
        bit_reverse_coset_to_circle_domain_order(col.as_mut_slice());
        //////////////////////////////////////////////////////////////

        CircleEvaluation::new(CanonicCoset::new(self.log_size).circle_domain(), col)
    }

    pub fn id(&self) -> PreProcessedColumnId {
        PreProcessedColumnId {
            id: format!("is_first_{}", self.log_size),
        }
    }
}

fn gen_trace(log_size: u32) -> Vec<CircleEvaluation<SimdBackend, M31, BitReversedOrder>> {
    // Create a table with random values
    let mut rng = rand::thread_rng();
    let sorted_values = (0..(1 << log_size)).map(|i| i).collect::<Vec<_>>();
    let mut unsorted_values = sorted_values.clone();
    unsorted_values.shuffle(&mut rng);

    let mut unsorted_col = BaseColumn::from_iter(unsorted_values.iter().map(|v| M31::from(*v)));
    let mut sorted_col = BaseColumn::from_iter(sorted_values.iter().map(|v| M31::from(*v)));

    // Convert table to trace polynomials
    let domain = CanonicCoset::new(log_size).circle_domain();

    ////////////////////////////////////////////////////////////////////
    // Convert the columns to bit-reversed circle domain order
    bit_reverse_coset_to_circle_domain_order(unsorted_col.as_mut_slice());
    bit_reverse_coset_to_circle_domain_order(sorted_col.as_mut_slice());
    ////////////////////////////////////////////////////////////////////

    vec![unsorted_col, sorted_col]
        .into_iter()
        .map(|col| CircleEvaluation::new(domain, col))
        .collect()
}
And voilà, we have successfully implemented the constraint. You can run it here.