fix!: fix GroupByExec so that it is composable with others

crates/proof-of-sql/src/sql/proof_plans/group_by_exec.rs

@@ -361,13 +361,13 @@ fn verify_group_by<S: Scalar>(
         g_in_star_eval * sel_in_eval * sum_in_fold_eval - g_out_star_eval * sum_out_bar_fold_eval,
     );
 
-    // g_in_star * g_in_fold - 1 = 0
+    // g_in_star * g_in_fold - input_ones = 0
     builder.produce_sumcheck_subpolynomial_evaluation(
         &SumcheckSubpolynomialType::Identity,
         g_in_star_eval * g_in_fold_eval - one_eval,
     );
 
-    // g_out_star * g_out_bar_fold - 1 = 0
+    // g_out_star * g_out_bar_fold - input_ones = 0
     builder.produce_sumcheck_subpolynomial_evaluation(
         &SumcheckSubpolynomialType::Identity,
         g_out_star_eval * g_out_bar_fold_eval - one_eval,
@@ -390,6 +390,7 @@ pub fn prove_group_by<'a, S: Scalar>(
     n: usize,
 ) {
     let m_out = count_out.len();
+    let input_ones = alloc.alloc_slice_fill_copy(n, true);
 
     // g_in_fold = alpha + sum beta^j * g_in[j]
     let g_in_fold = alloc.alloc_slice_fill_copy(n, alpha);
@@ -442,19 +443,19 @@ pub fn prove_group_by<'a, S: Scalar>(
         ],
     );
 
-    // g_in_star * g_in_fold - 1 = 0
+    // g_in_star * g_in_fold - input_ones = 0
     builder.produce_sumcheck_subpolynomial(
         SumcheckSubpolynomialType::Identity,
         vec![
             (
                 S::one(),
                 vec![Box::new(g_in_star as &[_]), Box::new(g_in_fold as &[_])],
             ),
-            (-S::one(), vec![]),
+            (-S::one(), vec![Box::new(input_ones as &[_])]),
         ],
     );
 
-    // g_out_star * g_out_bar_fold - 1 = 0
+    // g_out_star * g_out_bar_fold - input_ones = 0
     builder.produce_sumcheck_subpolynomial(
         SumcheckSubpolynomialType::Identity,
         vec![
@@ -465,7 +466,7 @@ pub fn prove_group_by<'a, S: Scalar>(
                     Box::new(g_out_bar_fold as &[_]),
                 ],
             ),
-            (-S::one(), vec![]),
+            (-S::one(), vec![Box::new(input_ones as &[_])]),
         ],
     );
 }