Function lookup_prove 

pub fn lookup_prove<const FULL_ROUNDS: usize, G, EFqSponge, EFrSponge, RNG>(
    input: LookupProofInput<G::ScalarField>,
    acc_init: G::ScalarField,
    srs: &SRS<G>,
    domain: EvaluationDomains<G::ScalarField>,
    fq_sponge: EFqSponge,
    constraint: &ELookup<G::ScalarField>,
    rng: &mut RNG,
) -> (Proof<FULL_ROUNDS, G>, G::ScalarField)
where
    G: KimchiCurve<FULL_ROUNDS>,
    EFqSponge: FqSponge<G::BaseField, G, G::ScalarField, FULL_ROUNDS> + Clone,
    EFrSponge: FrSponge<G::ScalarField> + From<&'static ArithmeticSpongeParams<G::ScalarField, FULL_ROUNDS>>,
    G::BaseField: PrimeField,
    RNG: RngCore + CryptoRng,

This prover takes one Public Input and one Public Output. It then proves that the sum 1/(beta + table) = PI - PO where the table term are term from fixed lookup or RAMLookup