Theorem: Groth16TypeI.is_sound
Proof state before:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
r : Fin n_wit → F
stmt : Groth16TypeI.Stmt
prover : Prover F Groth16TypeI
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
          (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                          (X Vars_γ * X Vars_x ^ ↑x *
                            (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
            -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
          -(C (prover.2 Proof_Idx.B SRS_Elements_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_Idx.C SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
              (C (prover.1 Proof_Idx.C SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
                (C (prover.1 Proof_Idx.C SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                            (fun x =>
                              C (prover.1 Proof_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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
eqnA :
  C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
      (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
        (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          ((List.map
                (fun x => C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                      (X Vars_γ * X Vars_x ^ ↑x *
                        (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                  (List.finRange (n_var - 1))).sum +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.A (SRS_Elements_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))))) =
    C (prover.2 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
      (C (prover.2 Proof_Idx.A SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ) +
        (C (prover.2 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          (List.map
              (fun x => C (prover.2 Proof_Idx.A (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
              (List.finRange n_var)).sum))
eqnB :
  C (prover.1 Proof_Idx.B SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
      (C (prover.1 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
        (C (prover.1 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          ((List.map
                (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_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_Idx.B (SRS_Elements_Idx.x_pow_times_t x)) *
                      (X Vars_γ * X Vars_x ^ ↑x *
                        (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                  (List.finRange (n_var - 1))).sum +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_Idx.B (SRS_Elements_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_Idx.B (SRS_Elements_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))))) =
    C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
      (C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ) +
        (C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          (List.map
              (fun x => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
              (List.finRange n_var)).sum))
eqnC :
  C (prover.1 Proof_Idx.C SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
      (C (prover.1 Proof_Idx.C SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
        (C (prover.1 Proof_Idx.C SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          ((List.map
                (fun x => C (prover.1 Proof_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) *
                      (X Vars_γ * X Vars_x ^ ↑x *
                        (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                  (List.finRange (n_var - 1))).sum +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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))))) =
    C (prover.2 Proof_Idx.C SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
      (C (prover.2 Proof_Idx.C SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ) +
        (C (prover.2 Proof_Idx.C SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          (List.map
              (fun x => C (prover.2 Proof_Idx.C (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
              (List.finRange n_var)).sum))
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum
Tactic:
simp only [mul_add, add_mul, List.sum_map_add] at eqn eqnA eqnB eqnC
Proof state after:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
r : Fin n_wit → F
stmt : Groth16TypeI.Stmt
prover : Prover F Groth16TypeI
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(List.map
                      (fun x =>
                        C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum +
              (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(List.map
                        (fun x =>
                          C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum +
                (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(List.map
                          (fun x =>
                            C (prover.2 Proof_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                            (List.finRange (n_var - 1))).sum *
                        -(List.map
                              (fun x =>
                                C (prover.2 Proof_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_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_Idx.A (SRS_Elements_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_Idx.B (SRS_Elements_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum)))))) +
            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
              (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                            (List.finRange (n_var - 1))).sum *
                        -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ))))))))) +
          (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
              (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum *
                      -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ))))))))) +
        (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
            -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
          (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
              -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
              ((List.map
                      (fun x =>
                        C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum *
                  -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                            (X Vars_γ * X Vars_x ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1))).sum *
                    -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                  ((List.map
                              (fun i =>
                                C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y i)) *
                                (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                            (List.finRange n_stmt)).sum *
                        -(C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                    ((List.map
                              (fun i =>
                                C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q i)) *
                                (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                            (List.finRange n_wit)).sum *
                        -(C (prover.2 Proof_Idx.B SRS_Elements_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_Idx.C SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              (X Vars_γ * X Vars_δ * X Vars_δ) +
            (C (prover.1 Proof_Idx.C SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                (X Vars_γ * X Vars_δ * X Vars_δ) +
              (C (prover.1 Proof_Idx.C SRS_Elements_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_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum *
                      (X Vars_γ * X Vars_δ * X Vars_δ) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_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
eqnA :
  C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
      (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
        (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          ((List.map
                (fun x => C (prover.1 Proof_Idx.A (SRS_Elements_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_Idx.A (SRS_Elements_Idx.x_pow_times_t x)) *
                      (X Vars_γ * X Vars_x ^ ↑x *
                        (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                  (List.finRange (n_var - 1))).sum +
              ((List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y i)) *
                            (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y i)) *
                            (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                        (List.finRange n_stmt)).sum +
                  (List.map
                      (fun i =>
                        C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y i)) *
                          (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                      (List.finRange n_stmt)).sum +
                ((List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q i)) *
                            (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                        (List.finRange n_wit)).sum +
                    (List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q i)) *
                            (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                        (List.finRange n_wit)).sum +
                  (List.map
                      (fun i =>
                        C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q i)) *
                          (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                      (List.finRange n_wit)).sum)))))) =
    C (prover.2 Proof_Idx.A SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
      (C (prover.2 Proof_Idx.A SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ) +
        (C (prover.2 Proof_Idx.A SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          (List.map
              (fun x => C (prover.2 Proof_Idx.A (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
              (List.finRange n_var)).sum))
eqnB :
  C (prover.1 Proof_Idx.B SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
      (C (prover.1 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
        (C (prover.1 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          ((List.map
                (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_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_Idx.B (SRS_Elements_Idx.x_pow_times_t x)) *
                      (X Vars_γ * X Vars_x ^ ↑x *
                        (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                  (List.finRange (n_var - 1))).sum +
              ((List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.B (SRS_Elements_Idx.y i)) *
                            (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.B (SRS_Elements_Idx.y i)) *
                            (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                        (List.finRange n_stmt)).sum +
                  (List.map
                      (fun i =>
                        C (prover.1 Proof_Idx.B (SRS_Elements_Idx.y i)) *
                          (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                      (List.finRange n_stmt)).sum +
                ((List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.B (SRS_Elements_Idx.q i)) *
                            (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                        (List.finRange n_wit)).sum +
                    (List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.B (SRS_Elements_Idx.q i)) *
                            (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                        (List.finRange n_wit)).sum +
                  (List.map
                      (fun i =>
                        C (prover.1 Proof_Idx.B (SRS_Elements_Idx.q i)) *
                          (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                      (List.finRange n_wit)).sum)))))) =
    C (prover.2 Proof_Idx.B SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
      (C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ) +
        (C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          (List.map
              (fun x => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
              (List.finRange n_var)).sum))
eqnC :
  C (prover.1 Proof_Idx.C SRS_Elements_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
      (C (prover.1 Proof_Idx.C SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
        (C (prover.1 Proof_Idx.C SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          ((List.map
                (fun x => C (prover.1 Proof_Idx.C (SRS_Elements_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_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) *
                      (X Vars_γ * X Vars_x ^ ↑x *
                        (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                  (List.finRange (n_var - 1))).sum +
              ((List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y i)) *
                            (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y i)) *
                            (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                        (List.finRange n_stmt)).sum +
                  (List.map
                      (fun i =>
                        C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y i)) *
                          (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                      (List.finRange n_stmt)).sum +
                ((List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) *
                            (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                        (List.finRange n_wit)).sum +
                    (List.map
                        (fun i =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) *
                            (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                        (List.finRange n_wit)).sum +
                  (List.map
                      (fun i =>
                        C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) *
                          (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                      (List.finRange n_wit)).sum)))))) =
    C (prover.2 Proof_Idx.C SRS_Elements_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
      (C (prover.2 Proof_Idx.C SRS_Elements_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ) +
        (C (prover.2 Proof_Idx.C SRS_Elements_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
          (List.map
              (fun x => C (prover.2 Proof_Idx.C (SRS_Elements_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
              (List.finRange n_var)).sum))
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum