diff --git a/machine/src/__internal/debug_builder.rs b/machine/src/__internal/debug_builder.rs
index 6cfcaa08..f689e15e 100644
--- a/machine/src/__internal/debug_builder.rs
+++ b/machine/src/__internal/debug_builder.rs
@@ -26,6 +26,10 @@ where
     type Var = SC::Val;
     type M = TwoRowMatrixView<'a, SC::Val>;
 
+    fn main(&self) -> Self::M {
+        self.main
+    }
+
     fn is_first_row(&self) -> Self::Expr {
         self.is_first_row
     }
@@ -42,10 +46,6 @@ where
         }
     }
 
-    fn main(&self) -> Self::M {
-        self.main
-    }
-
     fn assert_zero<I: Into<Self::Expr>>(&mut self, x: I) {
         assert_eq!(
             x.into(),
@@ -71,8 +71,8 @@ where
     SC: StarkConfig,
 {
     type EF = SC::Challenge;
-    type VarEF = SC::Challenge;
     type ExprEF = SC::Challenge;
+    type VarEF = SC::Challenge;
     type MP = TwoRowMatrixView<'a, SC::Challenge>;
 
     fn permutation(&self) -> Self::MP {
diff --git a/machine/src/__internal/folding_builder.rs b/machine/src/__internal/folding_builder.rs
index 85ecfea3..3b521da7 100644
--- a/machine/src/__internal/folding_builder.rs
+++ b/machine/src/__internal/folding_builder.rs
@@ -2,18 +2,18 @@ use crate::{Machine, ValidaAirBuilder};
 use p3_air::{AirBuilder, PairBuilder, PermutationAirBuilder, TwoRowMatrixView};
 use valida_machine::config::StarkConfig;
 
-pub struct ConstraintFolder<'a, M: Machine<SC::Val>, SC: StarkConfig> {
+pub struct ProverConstraintFolder<'a, M: Machine<SC::Val>, SC: StarkConfig> {
     pub(crate) machine: &'a M,
     pub(crate) main: TwoRowMatrixView<'a, SC::Val>,
     pub(crate) preprocessed: TwoRowMatrixView<'a, SC::Val>,
     pub(crate) perm: TwoRowMatrixView<'a, SC::Challenge>,
-    pub(crate) rand_elems: &'a [SC::Challenge],
+    pub(crate) perm_challenges: &'a [SC::Challenge],
     pub(crate) is_first_row: SC::Val,
     pub(crate) is_last_row: SC::Val,
     pub(crate) is_transition: SC::Val,
 }
 
-impl<'a, M, SC> AirBuilder for ConstraintFolder<'a, M, SC>
+impl<'a, M, SC> AirBuilder for ProverConstraintFolder<'a, M, SC>
 where
     M: Machine<SC::Val>,
     SC: StarkConfig,
@@ -23,6 +23,10 @@ where
     type Var = SC::Val; // TODO: PackedVal
     type M = TwoRowMatrixView<'a, SC::Val>; // TODO: PackedVal
 
+    fn main(&self) -> Self::M {
+        self.main
+    }
+
     fn is_first_row(&self) -> Self::Expr {
         self.is_first_row
     }
@@ -39,16 +43,12 @@ where
         }
     }
 
-    fn main(&self) -> Self::M {
-        self.main
-    }
-
     fn assert_zero<I: Into<Self::Expr>>(&mut self, _x: I) {
         // TODO
     }
 }
 
-impl<'a, M, SC> PairBuilder for ConstraintFolder<'a, M, SC>
+impl<'a, M, SC> PairBuilder for ProverConstraintFolder<'a, M, SC>
 where
     M: Machine<SC::Val>,
     SC: StarkConfig,
@@ -58,14 +58,14 @@ where
     }
 }
 
-impl<'a, M, SC> PermutationAirBuilder for ConstraintFolder<'a, M, SC>
+impl<'a, M, SC> PermutationAirBuilder for ProverConstraintFolder<'a, M, SC>
 where
     M: Machine<SC::Val>,
     SC: StarkConfig,
 {
     type EF = SC::Challenge;
-    type VarEF = SC::Challenge;
     type ExprEF = SC::Challenge;
+    type VarEF = SC::Challenge;
     type MP = TwoRowMatrixView<'a, SC::Challenge>; // TODO: packed challenge?
 
     fn permutation(&self) -> Self::MP {
@@ -74,11 +74,11 @@ where
 
     fn permutation_randomness(&self) -> &[Self::EF] {
         // TODO: implement
-        self.rand_elems
+        self.perm_challenges
     }
 }
 
-impl<'a, M, SC> ValidaAirBuilder for ConstraintFolder<'a, M, SC>
+impl<'a, M, SC> ValidaAirBuilder for ProverConstraintFolder<'a, M, SC>
 where
     M: Machine<SC::Val>,
     SC: StarkConfig,
diff --git a/machine/src/chip.rs b/machine/src/chip.rs
index 7719a36b..38cb71b1 100644
--- a/machine/src/chip.rs
+++ b/machine/src/chip.rs
@@ -1,5 +1,5 @@
 use crate::Machine;
-use crate::__internal::{ConstraintFolder, DebugConstraintBuilder};
+use crate::__internal::{DebugConstraintBuilder, ProverConstraintFolder};
 use alloc::vec;
 use alloc::vec::Vec;
 use valida_util::batch_multiplicative_inverse;
@@ -10,7 +10,7 @@ use p3_field::{AbstractExtensionField, AbstractField, ExtensionField, Field, Pow
 use p3_matrix::{dense::RowMajorMatrix, Matrix, MatrixRowSlices};
 
 pub trait Chip<M: Machine<SC::Val>, SC: StarkConfig>:
-    for<'a> Air<ConstraintFolder<'a, M, SC>> + for<'a> Air<DebugConstraintBuilder<'a, M, SC>>
+    for<'a> Air<ProverConstraintFolder<'a, M, SC>> + for<'a> Air<DebugConstraintBuilder<'a, M, SC>>
 {
     /// Generate the main trace for the chip given the provided machine.
     fn generate_trace(&self, machine: &M) -> RowMajorMatrix<SC::Val>;