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 :
  (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                    (List.finRange n_var)).sum +
                ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                          (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                            (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                              X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                            (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit x) +
                                X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit x) +
                              X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit x)))
                        (List.finRange n_wit)).sum)))))) *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
          -(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_α * (X Vars_γ * X Vars_δ * X Vars_β) +
        ((List.map
                (fun x =>
                  C (stmt x) *
                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                        X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                      X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                (List.finRange n_stmt)).sum *
            (X Vars_γ * X Vars_δ * X Vars_γ) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum +
                    ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt)).sum +
                        (List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit x) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit x) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit x)))
                            (List.finRange n_wit)).sum)))))) *
            (X Vars_γ * X Vars_δ * X Vars_δ))) =
    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 [mul_add, add_mul, List.sum_map_add] 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 :
  C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(List.map
                      (fun x =>
                        C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                          (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(List.map
                        (fun x =>
                          C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(List.map
                            (fun x =>
                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                            (List.finRange n_var)).sum +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                            (List.finRange (n_var - 1))).sum *
                        -(List.map
                              (fun x =>
                                C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                  (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                              (List.finRange n_var)).sum +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum)))))) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                            (List.finRange (n_var - 1))).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                (X Vars_γ * X Vars_δ * X Vars_δ))))))))) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                              (X Vars_γ * X Vars_δ * X Vars_γ))))))))) +
        (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
              ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                          (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                            (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                        (List.finRange (n_var - 1))).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                  ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                            (List.finRange n_stmt)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                    ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                            (List.finRange n_wit)).sum *
                        -(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_α * (X Vars_γ * X Vars_δ * X Vars_β) +
        ((List.map (fun i => C (stmt i) * (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X Vars_γ * X Vars_δ * X Vars_γ) +
              (List.map (fun i => C (stmt i) * (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X Vars_γ * X Vars_δ * X Vars_γ) +
            (List.map (fun i => C (stmt i) * (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                  (List.finRange n_stmt)).sum *
              (X Vars_γ * X Vars_δ * X Vars_γ) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              (X Vars_γ * X Vars_δ * X Vars_δ) +
            (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                (X Vars_γ * X Vars_δ * X Vars_δ) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  (X Vars_γ * X Vars_δ * X Vars_δ) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum *
                      (X Vars_γ * X Vars_δ * X Vars_δ) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          (X Vars_γ * X Vars_δ * X Vars_δ) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          (X Vars_γ * X Vars_δ * X Vars_δ)))))))))) =
    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