Theorem: Lipmaa.soundness (FormalSnarksProject/SNARKs/Lipmaa/Soundness.lean)

State before:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
r : Fin n_wit → F
stmt : Lipmaa.Stmt
prover : Prover F Lipmaa
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  -(X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 75 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          C
                            (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                              (List.map
                                  (fun x =>
                                    Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      Polynomial.X ^ ↑x)
                                  (List.finRange n_var)).sum)))))) +
              (-(X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 25 *
                        (X Vars.y ^ 5 *
                          (X Vars.y ^ 1 *
                            C
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                (List.map
                                    (fun x =>
                                      Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        Polynomial.X ^ ↑x)
                                    (List.finRange n_var)).sum)))))) +
                (-(X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              C
                                (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum)))))) +
                  (-(X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              C
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum))))) +
                    (-(X Vars.y ^ 5 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              C
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                          (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                      (List.finRange (n_var - 1))).sum *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum)))) +
                      (-(X Vars.y ^ 25 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  u_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                            -(X Vars.y ^ 75 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  v_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                          -(X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  C
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                          (List.finRange n_stmt)).sum *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                              Polynomial.X ^ ↑x)
                                          (List.finRange n_var)).sum)))) +
                        (-(X Vars.y ^ 25 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  u_wit x)
                                              (List.finRange n_wit)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                            -(X Vars.y ^ 75 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  v_wit x)
                                              (List.finRange n_wit)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                          -(X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  C
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                          (List.finRange n_wit)).sum *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                              Polynomial.X ^ ↑x)
                                          (List.finRange n_var)).sum)))))))))) +
            (-(X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 75 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 *
                            C
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
              (-(X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 25 *
                        (X Vars.y ^ 5 *
                          (X Vars.y ^ 1 *
                            (X Vars.y ^ 1 *
                              C
                                (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                (-(X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                C
                                  (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                  (-(X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                C
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                            Polynomial.X ^ ↑x)
                                        (List.finRange n_var)).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                    (-(X Vars.y ^ 5 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                C
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                            (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                        (List.finRange (n_var - 1))).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                      (-(X Vars.y ^ 25 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      (X Vars.y ^ 1 *
                                        C
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                    u_stmt x)
                                                (List.finRange n_stmt)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                            -(X Vars.y ^ 75 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      (X Vars.y ^ 1 *
                                        C
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                    v_stmt x)
                                                (List.finRange n_stmt)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                          -(X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    C
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                w_stmt x)
                                            (List.finRange n_stmt)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                        (-(X Vars.y ^ 25 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      (X Vars.y ^ 1 *
                                        C
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                    u_wit x)
                                                (List.finRange n_wit)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                            -(X Vars.y ^ 75 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 5 *
                                    (X Vars.y ^ 1 *
                                      (X Vars.y ^ 1 *
                                        C
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                    v_wit x)
                                                (List.finRange n_wit)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                          -(X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    C
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                w_wit x)
                                            (List.finRange n_wit)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))))))))) +
          (-(X Vars.y ^ 5 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 75 *
                    (X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 5 *
                          C
                            (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
            (-(X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 25 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            C
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
              (-(X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 5 *
                          (X Vars.y ^ 1 *
                            (X Vars.y ^ 5 *
                              C
                                (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                (-(X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 5 *
                          (X Vars.y ^ 1 *
                            (X Vars.y ^ 5 *
                              C
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                  (-(X Vars.y ^ 5 *
                        (X Vars.y ^ 5 *
                          (X Vars.y ^ 1 *
                            (X Vars.y ^ 5 *
                              C
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                          (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                      (List.finRange (n_var - 1))).sum *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                    (-(X Vars.y ^ 25 *
                              (X Vars.y ^ 1 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 1 *
                                    (X Vars.y ^ 5 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  u_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                          -(X Vars.y ^ 75 *
                              (X Vars.y ^ 1 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 1 *
                                    (X Vars.y ^ 5 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  v_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                        -(X Vars.y ^ 1 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 1 *
                                (X Vars.y ^ 5 *
                                  C
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                          (List.finRange n_stmt)).sum *
                                      Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                      (-(X Vars.y ^ 25 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 1 *
                                    (X Vars.y ^ 5 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  u_wit x)
                                              (List.finRange n_wit)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                          -(X Vars.y ^ 75 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 5 *
                                  (X Vars.y ^ 1 *
                                    (X Vars.y ^ 5 *
                                      C
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  v_wit x)
                                              (List.finRange n_wit)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                        -(X Vars.y ^ 5 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 1 *
                                (X Vars.y ^ 5 *
                                  C
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                          (List.finRange n_wit)).sum *
                                      Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))))))))) +
        (-(X Vars.y ^ 5 *
              (X Vars.y ^ 1 *
                (X Vars.y ^ 75 *
                  (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 25 *
                        C
                          (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
          (-(X Vars.y ^ 5 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 25 *
                    (X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 25 *
                          C
                            (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
            (-(X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 25 *
                            C
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
              (-(X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 25 *
                            C
                              ((List.map
                                    (fun x =>
                                      Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                        Polynomial.X ^ ↑x)
                                    (List.finRange n_var)).sum *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                (-(X Vars.y ^ 5 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 25 *
                            C
                              ((List.map
                                    (fun x =>
                                      Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                        (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                    (List.finRange (n_var - 1))).sum *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                  (-(X Vars.y ^ 25 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 25 *
                                    C
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                u_stmt x)
                                            (List.finRange n_stmt)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                        -(X Vars.y ^ 75 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 25 *
                                    C
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                v_stmt x)
                                            (List.finRange n_stmt)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                      -(X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 25 *
                                C
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                        (List.finRange n_stmt)).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                    (-(X Vars.y ^ 25 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 25 *
                                    C
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                u_wit x)
                                            (List.finRange n_wit)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                        -(X Vars.y ^ 75 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 25 *
                                    C
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                v_wit x)
                                            (List.finRange n_wit)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                      -(X Vars.y ^ 5 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 25 *
                                C
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                        (List.finRange n_wit)).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))))))))) +
      (X Vars.y ^ 5 * (X Vars.y ^ 1 * (X Vars.y ^ 75 * (X Vars.y ^ 5 * (X Vars.y ^ 1 * X Vars.y ^ 25)))) +
        (X Vars.y ^ 25 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        C (List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum)))) +
              X Vars.y ^ 75 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        C (List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum)))) +
            X Vars.y ^ 1 *
              (X Vars.y ^ 5 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 5 *
                    C (List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum))) +
          (X Vars.y ^ 5 *
              (X Vars.y ^ 1 *
                (X Vars.y ^ 75 *
                  (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 1 * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α))))))) +
            (X Vars.y ^ 5 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 25 *
                    (X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 1 * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β))))))) +
              (X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ))))))) +
                (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 *
                            C
                              (List.map
                                  (fun x =>
                                    Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                      Polynomial.X ^ ↑x)
                                  (List.finRange n_var)).sum)))) +
                  (X Vars.y ^ 5 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 *
                            C
                              (List.map
                                  (fun x =>
                                    Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                      (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                  (List.finRange (n_var - 1))).sum))) +
                    (X Vars.y ^ 25 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    C
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                                          (List.finRange n_stmt)).sum)))) +
                          X Vars.y ^ 75 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    C
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * v_stmt x)
                                          (List.finRange n_stmt)).sum)))) +
                        X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                C
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                      (List.finRange n_stmt)).sum))) +
                      (X Vars.y ^ 25 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    C
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * u_wit x)
                                          (List.finRange n_wit)).sum)))) +
                          X Vars.y ^ 75 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    C
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * v_wit x)
                                          (List.finRange n_wit)).sum)))) +
                        X Vars.y ^ 5 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                C
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                      (List.finRange n_wit)).sum)))))))))))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x =>
            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum
Tactic:
simp only [X, C_apply, monomial_mul, monomial_pow, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqn
State after:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
r : Fin n_wit → F
stmt : Lipmaa.Stmt
prover : Prover F Lipmaa
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  -(monomial
                    ((5 • fun₀ | Vars.y => 1) +
                      ((1 • fun₀ | Vars.y => 1) +
                        ((75 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                  (1 ^ 5 *
                    (1 ^ 1 *
                      (1 ^ 75 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                              (List.map
                                  (fun x =>
                                    Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      Polynomial.X ^ ↑x)
                                  (List.finRange n_var)).sum)))))) +
              (-(monomial
                      ((5 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((25 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 25 *
                          (1 ^ 5 *
                            (1 ^ 1 *
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                (List.map
                                    (fun x =>
                                      Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        Polynomial.X ^ ↑x)
                                    (List.finRange n_var)).sum)))))) +
                (-(monomial
                        ((5 • fun₀ | Vars.y => 1) +
                          ((1 • fun₀ | Vars.y => 1) +
                            ((1 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                      (1 ^ 5 *
                        (1 ^ 1 *
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum)))))) +
                  (-(monomial
                          ((5 • fun₀ | Vars.y => 1) +
                            ((1 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum))))) +
                    (-(monomial ((5 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))
                          (1 ^ 5 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                          (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                      (List.finRange (n_var - 1))).sum *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum)))) +
                      (-(monomial
                                  ((25 • fun₀ | Vars.y => 1) +
                                    ((1 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                                (1 ^ 25 *
                                  (1 ^ 1 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  u_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                            -(monomial
                                  ((75 • fun₀ | Vars.y => 1) +
                                    ((1 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                                (1 ^ 75 *
                                  (1 ^ 1 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  v_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                          -(monomial ((1 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))
                              (1 ^ 1 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                          (List.finRange n_stmt)).sum *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                              Polynomial.X ^ ↑x)
                                          (List.finRange n_var)).sum)))) +
                        (-(monomial
                                  ((25 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                                (1 ^ 25 *
                                  (1 ^ 5 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  u_wit x)
                                              (List.finRange n_wit)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                            -(monomial
                                  ((75 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                                (1 ^ 75 *
                                  (1 ^ 5 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  v_wit x)
                                              (List.finRange n_wit)).sum *
                                          (List.map
                                              (fun x =>
                                                Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                                  Polynomial.X ^ ↑x)
                                              (List.finRange n_var)).sum))))) +
                          -(monomial ((5 • fun₀ | Vars.y => 1) + ((5 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))
                              (1 ^ 5 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                          (List.finRange n_wit)).sum *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                              Polynomial.X ^ ↑x)
                                          (List.finRange n_var)).sum)))))))))) +
            (-(monomial
                    ((5 • fun₀ | Vars.y => 1) +
                      ((1 • fun₀ | Vars.y => 1) +
                        ((75 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))))
                  (1 ^ 5 *
                    (1 ^ 1 *
                      (1 ^ 75 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 1 *
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
              (-(monomial
                      ((5 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((25 • fun₀ | Vars.y => 1) +
                            ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))))
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 25 *
                          (1 ^ 5 *
                            (1 ^ 1 *
                              (1 ^ 1 *
                                (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                (-(monomial
                        ((5 • fun₀ | Vars.y => 1) +
                          ((1 • fun₀ | Vars.y => 1) +
                            ((1 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))))
                      (1 ^ 5 *
                        (1 ^ 1 *
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 1 *
                                  (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                  (-(monomial
                          ((5 • fun₀ | Vars.y => 1) +
                            ((1 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 1 *
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                            Polynomial.X ^ ↑x)
                                        (List.finRange n_var)).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                    (-(monomial
                            ((5 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                          (1 ^ 5 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 1 *
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                            (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                        (List.finRange (n_var - 1))).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                      (-(monomial
                                  ((25 • fun₀ | Vars.y => 1) +
                                    ((1 • fun₀ | Vars.y => 1) +
                                      ((5 • fun₀ | Vars.y => 1) +
                                        ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                                (1 ^ 25 *
                                  (1 ^ 1 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        (1 ^ 1 *
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                    u_stmt x)
                                                (List.finRange n_stmt)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                            -(monomial
                                  ((75 • fun₀ | Vars.y => 1) +
                                    ((1 • fun₀ | Vars.y => 1) +
                                      ((5 • fun₀ | Vars.y => 1) +
                                        ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                                (1 ^ 75 *
                                  (1 ^ 1 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        (1 ^ 1 *
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                    v_stmt x)
                                                (List.finRange n_stmt)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                          -(monomial
                                ((1 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                              (1 ^ 1 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 1 *
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                w_stmt x)
                                            (List.finRange n_stmt)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                        (-(monomial
                                  ((25 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) +
                                      ((5 • fun₀ | Vars.y => 1) +
                                        ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                                (1 ^ 25 *
                                  (1 ^ 5 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        (1 ^ 1 *
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                    u_wit x)
                                                (List.finRange n_wit)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                            -(monomial
                                  ((75 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) +
                                      ((5 • fun₀ | Vars.y => 1) +
                                        ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                                (1 ^ 75 *
                                  (1 ^ 5 *
                                    (1 ^ 5 *
                                      (1 ^ 1 *
                                        (1 ^ 1 *
                                          ((List.map
                                                (fun x =>
                                                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                    v_wit x)
                                                (List.finRange n_wit)).sum *
                                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                          -(monomial
                                ((5 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                              (1 ^ 5 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 1 *
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                w_wit x)
                                            (List.finRange n_wit)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))))))))) +
          (-(monomial
                  ((5 • fun₀ | Vars.y => 1) +
                    ((1 • fun₀ | Vars.y => 1) +
                      ((75 • fun₀ | Vars.y => 1) +
                        ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))))
                (1 ^ 5 *
                  (1 ^ 1 *
                    (1 ^ 75 *
                      (1 ^ 5 *
                        (1 ^ 1 *
                          (1 ^ 5 *
                            (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
            (-(monomial
                    ((5 • fun₀ | Vars.y => 1) +
                      ((1 • fun₀ | Vars.y => 1) +
                        ((25 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))))
                  (1 ^ 5 *
                    (1 ^ 1 *
                      (1 ^ 25 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
              (-(monomial
                      ((5 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((1 • fun₀ | Vars.y => 1) +
                            ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))))
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 1 *
                          (1 ^ 5 *
                            (1 ^ 1 *
                              (1 ^ 5 *
                                (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                (-(monomial
                        ((5 • fun₀ | Vars.y => 1) +
                          ((1 • fun₀ | Vars.y => 1) +
                            ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                      (1 ^ 5 *
                        (1 ^ 1 *
                          (1 ^ 5 *
                            (1 ^ 1 *
                              (1 ^ 5 *
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                          Polynomial.X ^ ↑x)
                                      (List.finRange n_var)).sum *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                  (-(monomial
                          ((5 • fun₀ | Vars.y => 1) +
                            ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))
                        (1 ^ 5 *
                          (1 ^ 5 *
                            (1 ^ 1 *
                              (1 ^ 5 *
                                ((List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                          (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                      (List.finRange (n_var - 1))).sum *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                    (-(monomial
                                ((25 • fun₀ | Vars.y => 1) +
                                  ((1 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                              (1 ^ 25 *
                                (1 ^ 1 *
                                  (1 ^ 5 *
                                    (1 ^ 1 *
                                      (1 ^ 5 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  u_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                          -(monomial
                                ((75 • fun₀ | Vars.y => 1) +
                                  ((1 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                              (1 ^ 75 *
                                (1 ^ 1 *
                                  (1 ^ 5 *
                                    (1 ^ 1 *
                                      (1 ^ 5 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                  v_stmt x)
                                              (List.finRange n_stmt)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                        -(monomial
                              ((1 • fun₀ | Vars.y => 1) +
                                ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))
                            (1 ^ 1 *
                              (1 ^ 5 *
                                (1 ^ 1 *
                                  (1 ^ 5 *
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                          (List.finRange n_stmt)).sum *
                                      Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                      (-(monomial
                                ((25 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                              (1 ^ 25 *
                                (1 ^ 5 *
                                  (1 ^ 5 *
                                    (1 ^ 1 *
                                      (1 ^ 5 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  u_wit x)
                                              (List.finRange n_wit)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                          -(monomial
                                ((75 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) +
                                    ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                              (1 ^ 75 *
                                (1 ^ 5 *
                                  (1 ^ 5 *
                                    (1 ^ 1 *
                                      (1 ^ 5 *
                                        ((List.map
                                              (fun x =>
                                                Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                  v_wit x)
                                              (List.finRange n_wit)).sum *
                                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                        -(monomial
                              ((5 • fun₀ | Vars.y => 1) +
                                ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))
                            (1 ^ 5 *
                              (1 ^ 5 *
                                (1 ^ 1 *
                                  (1 ^ 5 *
                                    ((List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                          (List.finRange n_wit)).sum *
                                      Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))))))))) +
        (-(monomial
                ((5 • fun₀ | Vars.y => 1) +
                  ((1 • fun₀ | Vars.y => 1) +
                    ((75 • fun₀ | Vars.y => 1) +
                      ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))))
              (1 ^ 5 *
                (1 ^ 1 *
                  (1 ^ 75 *
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 25 *
                          (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
          (-(monomial
                  ((5 • fun₀ | Vars.y => 1) +
                    ((1 • fun₀ | Vars.y => 1) +
                      ((25 • fun₀ | Vars.y => 1) +
                        ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))))
                (1 ^ 5 *
                  (1 ^ 1 *
                    (1 ^ 25 *
                      (1 ^ 5 *
                        (1 ^ 1 *
                          (1 ^ 25 *
                            (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
            (-(monomial
                    ((5 • fun₀ | Vars.y => 1) +
                      ((1 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))))
                  (1 ^ 5 *
                    (1 ^ 1 *
                      (1 ^ 1 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 25 *
                              (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
              (-(monomial
                      ((5 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1)))))
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 25 *
                              ((List.map
                                    (fun x =>
                                      Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                        Polynomial.X ^ ↑x)
                                    (List.finRange n_var)).sum *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                (-(monomial
                        ((5 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))
                      (1 ^ 5 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 25 *
                              ((List.map
                                    (fun x =>
                                      Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                        (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                    (List.finRange (n_var - 1))).sum *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                  (-(monomial
                              ((25 • fun₀ | Vars.y => 1) +
                                ((1 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1)))))
                            (1 ^ 25 *
                              (1 ^ 1 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 25 *
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                u_stmt x)
                                            (List.finRange n_stmt)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                        -(monomial
                              ((75 • fun₀ | Vars.y => 1) +
                                ((1 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1)))))
                            (1 ^ 75 *
                              (1 ^ 1 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 25 *
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                                v_stmt x)
                                            (List.finRange n_stmt)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                      -(monomial
                            ((1 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 25 *
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                        (List.finRange n_stmt)).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                    (-(monomial
                              ((25 • fun₀ | Vars.y => 1) +
                                ((5 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1)))))
                            (1 ^ 25 *
                              (1 ^ 5 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 25 *
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                u_wit x)
                                            (List.finRange n_wit)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                        -(monomial
                              ((75 • fun₀ | Vars.y => 1) +
                                ((5 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1)))))
                            (1 ^ 75 *
                              (1 ^ 5 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 25 *
                                      ((List.map
                                            (fun x =>
                                              Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                                v_wit x)
                                            (List.finRange n_wit)).sum *
                                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                      -(monomial
                            ((5 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))
                          (1 ^ 5 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 25 *
                                  ((List.map
                                        (fun x =>
                                          Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                        (List.finRange n_wit)).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))))))))) +
      ((monomial
            ((5 • fun₀ | Vars.y => 1) +
              ((1 • fun₀ | Vars.y => 1) +
                ((75 • fun₀ | Vars.y => 1) +
                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 25 • fun₀ | Vars.y => 1))))))
          (1 ^ 5 * (1 ^ 1 * (1 ^ 75 * (1 ^ 5 * (1 ^ 1 * 1 ^ 25))))) +
        ((monomial
                  ((25 • fun₀ | Vars.y => 1) +
                    ((1 • fun₀ | Vars.y => 1) +
                      ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                (1 ^ 25 *
                  (1 ^ 1 *
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 5 *
                          (List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum))))) +
              (monomial
                  ((75 • fun₀ | Vars.y => 1) +
                    ((1 • fun₀ | Vars.y => 1) +
                      ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1)))))
                (1 ^ 75 *
                  (1 ^ 1 *
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 5 *
                          (List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum))))) +
            (monomial
                ((1 • fun₀ | Vars.y => 1) +
                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 5 • fun₀ | Vars.y => 1))))
              (1 ^ 1 *
                (1 ^ 5 *
                  (1 ^ 1 *
                    (1 ^ 5 * (List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum)))) +
          ((monomial
                ((5 • fun₀ | Vars.y => 1) +
                  ((1 • fun₀ | Vars.y => 1) +
                    ((75 • fun₀ | Vars.y => 1) +
                      ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))))
              (1 ^ 5 *
                (1 ^ 1 *
                  (1 ^ 75 *
                    (1 ^ 5 * (1 ^ 1 * (1 ^ 1 * Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α))))))) +
            ((monomial
                  ((5 • fun₀ | Vars.y => 1) +
                    ((1 • fun₀ | Vars.y => 1) +
                      ((25 • fun₀ | Vars.y => 1) +
                        ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))))
                (1 ^ 5 *
                  (1 ^ 1 *
                    (1 ^ 25 *
                      (1 ^ 5 * (1 ^ 1 * (1 ^ 1 * Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β))))))) +
              ((monomial
                    ((5 • fun₀ | Vars.y => 1) +
                      ((1 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))))
                  (1 ^ 5 *
                    (1 ^ 1 *
                      (1 ^ 1 *
                        (1 ^ 5 * (1 ^ 1 * (1 ^ 1 * Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ))))))) +
                ((monomial
                      ((5 • fun₀ | Vars.y => 1) +
                        ((1 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                    (1 ^ 5 *
                      (1 ^ 1 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 1 *
                              (List.map
                                  (fun x =>
                                    Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                      Polynomial.X ^ ↑x)
                                  (List.finRange n_var)).sum))))) +
                  ((monomial
                        ((5 • fun₀ | Vars.y => 1) +
                          ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                      (1 ^ 5 *
                        (1 ^ 5 *
                          (1 ^ 1 *
                            (1 ^ 1 *
                              (List.map
                                  (fun x =>
                                    Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                      (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
                                  (List.finRange (n_var - 1))).sum)))) +
                    ((monomial
                              ((25 • fun₀ | Vars.y => 1) +
                                ((1 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                            (1 ^ 25 *
                              (1 ^ 1 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 1 *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                                          (List.finRange n_stmt)).sum))))) +
                          (monomial
                              ((75 • fun₀ | Vars.y => 1) +
                                ((1 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                            (1 ^ 75 *
                              (1 ^ 1 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 1 *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * v_stmt x)
                                          (List.finRange n_stmt)).sum))))) +
                        (monomial
                            ((1 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                          (1 ^ 1 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 1 *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                      (List.finRange n_stmt)).sum)))) +
                      ((monomial
                              ((25 • fun₀ | Vars.y => 1) +
                                ((5 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                            (1 ^ 25 *
                              (1 ^ 5 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 1 *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * u_wit x)
                                          (List.finRange n_wit)).sum))))) +
                          (monomial
                              ((75 • fun₀ | Vars.y => 1) +
                                ((5 • fun₀ | Vars.y => 1) +
                                  ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1)))))
                            (1 ^ 75 *
                              (1 ^ 5 *
                                (1 ^ 5 *
                                  (1 ^ 1 *
                                    (1 ^ 1 *
                                      (List.map
                                          (fun x =>
                                            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * v_wit x)
                                          (List.finRange n_wit)).sum))))) +
                        (monomial
                            ((5 • fun₀ | Vars.y => 1) +
                              ((5 • fun₀ | Vars.y => 1) + ((1 • fun₀ | Vars.y => 1) + 1 • fun₀ | Vars.y => 1))))
                          (1 ^ 5 *
                            (1 ^ 5 *
                              (1 ^ 1 *
                                (1 ^ 1 *
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                      (List.finRange n_wit)).sum))))))))))))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x =>
            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum