diff --git a/src/lib.rs b/src/lib.rs
index d7a846e..c5f5a26 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -41,7 +41,7 @@ impl<F: PrimeField, P: UVPolynomial<F>> FRI<F, P> {
         );
     }
 
-    pub fn prove<R: Rng>(rng: &mut R, p: &P) -> (Vec<F>, [F; 2]) {
+    pub fn prove<R: Rng>(rng: &mut R, p: &P) -> (Vec<F>, Vec<F>, [F; 2]) {
         // f_0(x) = fL_0(x^2) + x fR_0(x^2)
         let mut f_i1 = p.clone();
 
@@ -59,15 +59,84 @@ impl<F: PrimeField, P: UVPolynomial<F>> FRI<F, P> {
             let aux = DensePolynomial::from_coefficients_slice(fR_i.coeffs());
             f_i1 = fL_i.clone() + P::from_coefficients_slice(aux.mul(alpha_i).coeffs());
             f_is.push(f_i1.clone());
+
+            // commit to f_{i+1}(x) = fL_i(x) + alpha_i * fR_i(x)
+            let (cm_i, mt_i) = MT::commit(f_i1.coeffs());
+            commitments.push(cm_i);
+            mts.push(mt_i);
         }
         let (fL_i, fR_i) = Self::split(&f_i1);
         let constant_fL_l: F = fL_i.coeffs()[0].clone();
         let constant_fR_l: F = fR_i.coeffs()[0].clone();
 
-        // TODO evaluate f_i(z^{2^i})
-        let evals: Vec<F> = Vec::new();
+        // TODO this will be a hash from the transcript
+        // V sets rand z \in \mathbb{F} challenge
+        let z = F::from(10_u64);
+
+        let mut evals: Vec<F> = Vec::new();
+        // TODO this will be done inside the prev loop, now it is here just for clarity
+        // evaluate f_i(z^{2^i})
+        for i in 0..f_is.len() {
+            // TODO check usage of .pow(u64)
+            let z_2i = z.pow([2_u64.pow(i as u32)]); // z^{2^i}
+            let neg_z_2i = z_2i.neg();
+            let eval_i = f_is[i].evaluate(&z_2i);
+            evals.push(eval_i);
+            let eval_i = f_is[i].evaluate(&neg_z_2i);
+            evals.push(eval_i);
+        }
+
+        // TODO return also the commitment_proofs
+        // return: Comm(f_i(x)), f_i(+-z^{2^i}), constant values {f_l^L, f_l^R}
+        (commitments, evals, [constant_fL_l, constant_fR_l])
+    }
+
+    pub fn verify(commitments: Vec<F>, evals: Vec<F>, constants: [F; 2]) -> bool {
+        let z = F::from(10_u64); // TODO this will be a hash from the transcript
+
+        // TODO check commitments.len()==evals.len()/2
+
+        for i in (0..evals.len()).step_by(2) {
+            let alpha_i = F::from(3_u64); // TODO: WIP, defined by Verifier (well, hash transcript)
+
+            let z_2i = z.pow([2_u64.pow((i as u32) / 2)]); // z^{2^i}
+                                                           // take f_i(z^2) from evals
+            let fi_z = evals[i];
+            let neg_fi_z = evals[i + 1];
+            // compute f_i^L(z^2), f_i^R(z^2) from the linear combination
+            let L = (fi_z + neg_fi_z) * F::from(2_u32).inverse().unwrap();
+            let R = (fi_z - neg_fi_z) * (F::from(2_u32) * z_2i).inverse().unwrap();
+
+            // compute f_{i+1}(z^2) = f_i^L(z^2) + a_i f_i^R(z^2)
+            let next_fi_z2 = L + alpha_i * R;
+
+            // check: obtained f_{i+1}(z^2) == evals.f_{i+1}(z^2) (=evals[i+2])
+            if i < evals.len() - 2 {
+                if next_fi_z2 != evals[i + 2] {
+                    println!("\nerr, i={:?}", i);
+                    println!("  next_fi^z2 {:?}", next_fi_z2.to_string());
+                    println!("  e[i] {:?}", evals[i + 2].to_string());
+                    panic!("should f_i+1(z^2) == evals.f_i+1(z^2) (=evals[i+2])");
+                }
+            }
+
+            // check commitment opening
+            // TODO
+
+            // last iteration, check constant values equal to the obtained f_i^L(z^{2^i}),
+            // f_i^R(z^{2^i})
+            if i == evals.len() - 2 {
+                if L != constants[0] {
+                    panic!("constant L not equal");
+                }
+                if R != constants[1] {
+                    println!("R {:?}\n  {:?}", R.to_string(), constants[1].to_string());
+                    panic!("constant R not equal");
+                }
+            }
+        }
 
-        (evals, [constant_fL_l, constant_fR_l])
+        true
     }
 }
 
@@ -97,4 +166,24 @@ mod tests {
             pL.evaluate(&z.square()) + z * pR.evaluate(&z.square())
         );
     }
+
+    #[test]
+    fn test_prove() {
+        let mut rng = ark_std::test_rng();
+
+        let deg = 15;
+        let p = DensePolynomial::<Fr>::rand(deg, &mut rng);
+        assert_eq!(p.degree(), deg);
+        // println!("p {:?}", p);
+
+        type FRIC = FRI<Fr, DensePolynomial<Fr>>;
+        // prover
+        let (commitments, evals, constvals) = FRIC::prove(&mut rng, &p);
+        // commitments contains the commitments to each f_0, f_1, ..., f_n, with n=log2(d)
+        assert_eq!(commitments.len(), log2(p.coeffs().len()) as usize - 1);
+        assert_eq!(evals.len(), 2 * log2(p.coeffs().len()) as usize);
+
+        let v = FRIC::verify(commitments, evals, constvals);
+        assert!(v);
+    }
 }