Theorem: Groth16TypeIII.is_sound_quotient_version
Proof 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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  -(X Vars.γ *
                  (X Vars.δ *
                    (X Vars.α *
                      (X Vars.γ *
                        (X Vars.δ *
                          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.γ *
                    (X Vars.δ *
                      (X Vars.β *
                        (X Vars.γ *
                          (X Vars.δ *
                            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.γ *
                      (X Vars.δ *
                        (X Vars.δ *
                          (X Vars.γ *
                            (X Vars.δ *
                              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.γ *
                        (X Vars.δ *
                          (X Vars.γ *
                            (X Vars.δ *
                              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.γ *
                          (X Vars.γ *
                            (X Vars.δ *
                              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 * t))
                                      (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.β *
                                (X Vars.δ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      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.α *
                                (X Vars.δ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      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.δ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  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.β *
                                (X Vars.γ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      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.α *
                                (X Vars.γ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      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.γ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  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.γ *
                  (X Vars.δ *
                    (X Vars.α *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.δ *
                            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.γ *
                    (X Vars.δ *
                      (X Vars.β *
                        (X Vars.γ *
                          (X Vars.δ *
                            (X Vars.δ *
                              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.γ *
                      (X Vars.δ *
                        (X Vars.δ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.δ *
                                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.γ *
                        (X Vars.δ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.δ *
                                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.γ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.δ *
                                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 * t))
                                        (List.finRange (n_var - 1))).sum *
                                    Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                      (-(X Vars.β *
                                (X Vars.δ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      (X Vars.δ *
                                        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.α *
                                (X Vars.δ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      (X Vars.δ *
                                        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.δ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.δ *
                                    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.β *
                                (X Vars.γ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      (X Vars.δ *
                                        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.α *
                                (X Vars.γ *
                                  (X Vars.γ *
                                    (X Vars.δ *
                                      (X Vars.δ *
                                        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.γ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.δ *
                                    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.γ *
                (X Vars.δ *
                  (X Vars.α *
                    (X Vars.γ *
                      (X Vars.δ *
                        (X Vars.γ *
                          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.γ *
                  (X Vars.δ *
                    (X Vars.β *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.γ *
                            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.γ *
                    (X Vars.δ *
                      (X Vars.δ *
                        (X Vars.γ *
                          (X Vars.δ *
                            (X Vars.γ *
                              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.γ *
                      (X Vars.δ *
                        (X Vars.γ *
                          (X Vars.δ *
                            (X Vars.γ *
                              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.γ *
                        (X Vars.γ *
                          (X Vars.δ *
                            (X Vars.γ *
                              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 * t))
                                      (List.finRange (n_var - 1))).sum *
                                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                    (-(X Vars.β *
                              (X Vars.δ *
                                (X Vars.γ *
                                  (X Vars.δ *
                                    (X Vars.γ *
                                      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.α *
                              (X Vars.δ *
                                (X Vars.γ *
                                  (X Vars.δ *
                                    (X Vars.γ *
                                      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.δ *
                            (X Vars.γ *
                              (X Vars.δ *
                                (X Vars.γ *
                                  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.β *
                              (X Vars.γ *
                                (X Vars.γ *
                                  (X Vars.δ *
                                    (X Vars.γ *
                                      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.α *
                              (X Vars.γ *
                                (X Vars.γ *
                                  (X Vars.δ *
                                    (X Vars.γ *
                                      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.γ *
                            (X Vars.γ *
                              (X Vars.δ *
                                (X Vars.γ *
                                  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.γ *
              (X Vars.δ *
                (X Vars.α *
                  (X Vars.γ *
                    (X Vars.δ *
                      (X Vars.β *
                        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.γ *
                (X Vars.δ *
                  (X Vars.β *
                    (X Vars.γ *
                      (X Vars.δ *
                        (X Vars.β *
                          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.γ *
                  (X Vars.δ *
                    (X Vars.δ *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.β *
                            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.γ *
                    (X Vars.δ *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.β *
                            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.γ *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.β *
                            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 * t))
                                    (List.finRange (n_var - 1))).sum *
                                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                  (-(X Vars.β *
                            (X Vars.δ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.β *
                                    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.α *
                            (X Vars.δ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.β *
                                    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.δ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.β *
                                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.β *
                            (X Vars.γ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.β *
                                    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.α *
                            (X Vars.γ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.β *
                                    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.γ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.β *
                                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.γ * (X Vars.δ * (X Vars.α * (X Vars.γ * (X Vars.δ * X Vars.β)))) +
        (X Vars.β *
                (X Vars.δ *
                  (X Vars.γ *
                    (X Vars.δ *
                      (X Vars.γ *
                        C (List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum)))) +
              X Vars.α *
                (X Vars.δ *
                  (X Vars.γ *
                    (X Vars.δ *
                      (X Vars.γ *
                        C (List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum)))) +
            X Vars.δ *
              (X Vars.γ *
                (X Vars.δ *
                  (X Vars.γ * C (List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum))) +
          (X Vars.γ *
              (X Vars.δ *
                (X Vars.α *
                  (X Vars.γ *
                    (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α))))))) +
            (X Vars.γ *
                (X Vars.δ *
                  (X Vars.β *
                    (X Vars.γ *
                      (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β))))))) +
              (X Vars.γ *
                  (X Vars.δ *
                    (X Vars.δ *
                      (X Vars.γ *
                        (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ))))))) +
                (X Vars.γ *
                    (X Vars.δ *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.δ *
                            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.γ *
                      (X Vars.γ *
                        (X Vars.δ *
                          (X Vars.δ *
                            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 * t))
                                  (List.finRange (n_var - 1))).sum))) +
                    (X Vars.β *
                            (X Vars.δ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.δ *
                                    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.α *
                            (X Vars.δ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.δ *
                                    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.δ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.δ *
                                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.β *
                            (X Vars.γ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.δ *
                                    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.α *
                            (X Vars.γ *
                              (X Vars.γ *
                                (X Vars.δ *
                                  (X Vars.δ *
                                    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.γ *
                          (X Vars.γ *
                            (X Vars.δ *
                              (X Vars.δ *
                                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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0
Tactic:
simp only [X, C_apply, monomial_mul, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqn
Proof 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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  -(monomial
                    ((fun₀ | Vars.γ => 1) +
                      ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.α => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                      ((fun₀ | Vars.γ => 1) +
                        ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                        ((fun₀ | Vars.γ => 1) +
                          ((fun₀ | Vars.δ => 1) +
                            ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                          ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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 ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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 * t))
                                (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
                                  ((fun₀ | Vars.β => 1) +
                                    ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                                  ((fun₀ | Vars.α => 1) +
                                    ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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 ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                                  ((fun₀ | Vars.β => 1) +
                                    ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                                  ((fun₀ | Vars.α => 1) +
                                    ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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 ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.δ => 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
                    ((fun₀ | Vars.γ => 1) +
                      ((fun₀ | Vars.δ => 1) +
                        ((fun₀ | Vars.α => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                      ((fun₀ | Vars.γ => 1) +
                        ((fun₀ | Vars.δ => 1) +
                          ((fun₀ | Vars.β => 1) +
                            ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                        ((fun₀ | Vars.γ => 1) +
                          ((fun₀ | Vars.δ => 1) +
                            ((fun₀ | Vars.δ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                          ((fun₀ | Vars.γ => 1) +
                            ((fun₀ | Vars.δ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                            ((fun₀ | Vars.γ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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 * t))
                                (List.finRange (n_var - 1))).sum *
                            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
                      (-(monomial
                                  ((fun₀ | Vars.β => 1) +
                                    ((fun₀ | Vars.δ => 1) +
                                      ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                                  ((fun₀ | Vars.α => 1) +
                                    ((fun₀ | Vars.δ => 1) +
                                      ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                                ((fun₀ | Vars.δ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                                  ((fun₀ | Vars.β => 1) +
                                    ((fun₀ | Vars.γ => 1) +
                                      ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                                  ((fun₀ | Vars.α => 1) +
                                    ((fun₀ | Vars.γ => 1) +
                                      ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                  ((fun₀ | Vars.γ => 1) +
                    ((fun₀ | Vars.δ => 1) +
                      ((fun₀ | Vars.α => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
            (-(monomial
                    ((fun₀ | Vars.γ => 1) +
                      ((fun₀ | Vars.δ => 1) +
                        ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
              (-(monomial
                      ((fun₀ | Vars.γ => 1) +
                        ((fun₀ | Vars.δ => 1) +
                          ((fun₀ | Vars.δ => 1) +
                            ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                (-(monomial
                        ((fun₀ | Vars.γ => 1) +
                          ((fun₀ | Vars.δ => 1) +
                            ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                  (-(monomial
                          ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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 * t))
                              (List.finRange (n_var - 1))).sum *
                          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
                    (-(monomial
                                ((fun₀ | Vars.β => 1) +
                                  ((fun₀ | Vars.δ => 1) +
                                    ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                          -(monomial
                                ((fun₀ | Vars.α => 1) +
                                  ((fun₀ | Vars.δ => 1) +
                                    ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                        -(monomial
                              ((fun₀ | Vars.δ => 1) +
                                ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                      (-(monomial
                                ((fun₀ | Vars.β => 1) +
                                  ((fun₀ | Vars.γ => 1) +
                                    ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                          -(monomial
                                ((fun₀ | Vars.α => 1) +
                                  ((fun₀ | Vars.γ => 1) +
                                    ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ)) +
                        -(monomial
                              ((fun₀ | Vars.γ => 1) +
                                ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 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.γ))))))))) +
        (-(monomial
                ((fun₀ | Vars.γ => 1) +
                  ((fun₀ | Vars.δ => 1) +
                    ((fun₀ | Vars.α => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
          (-(monomial
                  ((fun₀ | Vars.γ => 1) +
                    ((fun₀ | Vars.δ => 1) +
                      ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
            (-(monomial
                    ((fun₀ | Vars.γ => 1) +
                      ((fun₀ | Vars.δ => 1) +
                        ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
              (-(monomial
                      ((fun₀ | Vars.γ => 1) +
                        ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
                (-(monomial
                        ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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 * t))
                            (List.finRange (n_var - 1))).sum *
                        Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
                  (-(monomial
                              ((fun₀ | Vars.β => 1) +
                                ((fun₀ | Vars.δ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
                        -(monomial
                              ((fun₀ | Vars.α => 1) +
                                ((fun₀ | Vars.δ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
                      -(monomial
                            ((fun₀ | Vars.δ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
                    (-(monomial
                              ((fun₀ | Vars.β => 1) +
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
                        -(monomial
                              ((fun₀ | Vars.α => 1) +
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β)) +
                      -(monomial
                            ((fun₀ | Vars.γ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 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.β))))))))) +
      ((monomial
            ((fun₀ | Vars.γ => 1) +
              ((fun₀ | Vars.δ => 1) +
                ((fun₀ | Vars.α => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.β => 1))))))
          1 +
        ((monomial
                  ((fun₀ | Vars.β => 1) +
                    ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 1)))))
                (List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum +
              (monomial
                  ((fun₀ | Vars.α => 1) +
                    ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 1)))))
                (List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum +
            (monomial ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.γ => 1))))
              (List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum +
          ((monomial
                ((fun₀ | Vars.γ => 1) +
                  ((fun₀ | Vars.δ => 1) +
                    ((fun₀ | Vars.α => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 1))))))
              (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)) +
            ((monomial
                  ((fun₀ | Vars.γ => 1) +
                    ((fun₀ | Vars.δ => 1) +
                      ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 1))))))
                (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)) +
              ((monomial
                    ((fun₀ | Vars.γ => 1) +
                      ((fun₀ | Vars.δ => 1) +
                        ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 1))))))
                  (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)) +
                ((monomial
                      ((fun₀ | Vars.γ => 1) +
                        ((fun₀ | Vars.δ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                        ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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 * t))
                          (List.finRange (n_var - 1))).sum +
                    ((monomial
                              ((fun₀ | Vars.β => 1) +
                                ((fun₀ | Vars.δ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                              ((fun₀ | Vars.α => 1) +
                                ((fun₀ | Vars.δ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                            ((fun₀ | Vars.δ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                              ((fun₀ | Vars.β => 1) +
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                              ((fun₀ | Vars.α => 1) +
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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
                            ((fun₀ | Vars.γ => 1) +
                              ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.δ => 1) + fun₀ | Vars.δ => 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0