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) *
                -(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
Tactic:
simp only [
  mul_assoc,
  mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,
  mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),
  neg_mul, mul_neg,
  List.sum_map_mul_right, List.sum_map_mul_left] 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 :
  -(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_α *
                      (X Vars_γ *
                        (X Vars_δ *
                          (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) *
                            (List.map (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                (List.finRange n_var)).sum)))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_β *
                        (X Vars_γ *
                          (X Vars_δ *
                            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) *
                              (List.map (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                  (List.finRange n_var)).sum)))))) +
                (-(X Vars_γ *
                      (X Vars_δ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) *
                                (List.map (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum)))))) +
                  (-(X Vars_γ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              ((List.map (fun b => C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum *
                                (List.map (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum))))) +
                    (-(X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              ((List.map
                                    (fun b =>
                                      C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t b)) *
                                        (X Vars_x ^ ↑b *
                                          (to_MvPolynomial_Option Vars)
                                            (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                                    (List.finRange (n_var - 1))).sum *
                                (List.map (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum)))) +
                      (-(X Vars_β *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (u_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                            -(X Vars_α *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (v_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                          -(X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (w_stmt b))
                                        (List.finRange n_stmt)).sum *
                                    (List.map
                                        (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                        (List.finRange n_var)).sum)))) +
                        (-(X Vars_β *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (u_wit b))
                                            (List.finRange n_wit)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                            -(X Vars_α *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (v_wit b))
                                            (List.finRange n_wit)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                          -(X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (w_wit b))
                                        (List.finRange n_wit)).sum *
                                    (List.map
                                        (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                        (List.finRange n_var)).sum)))))))))) +
            (-(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_α *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_δ *
                            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) *
                              C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ)))))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_β *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_δ *
                              (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) *
                                C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ)))))))) +
                (-(X Vars_γ *
                      (X Vars_δ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) *
                                  C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ)))))))) +
                  (-(X Vars_γ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                ((List.map
                                      (fun b => C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                      (List.finRange n_var)).sum *
                                  C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ))))))) +
                    (-(X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t b)) *
                                          (X Vars_x ^ ↑b *
                                            (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_β *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                                  (to_MvPolynomial_Option Vars) (u_stmt b))
                                              (List.finRange n_stmt)).sum *
                                          C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ))))))) +
                            -(X Vars_α *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                                  (to_MvPolynomial_Option Vars) (v_stmt b))
                                              (List.finRange n_stmt)).sum *
                                          C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ))))))) +
                          -(X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                              (to_MvPolynomial_Option Vars) (w_stmt b))
                                          (List.finRange n_stmt)).sum *
                                      C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ)))))) +
                        (-(X Vars_β *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                                  (to_MvPolynomial_Option Vars) (u_wit b))
                                              (List.finRange n_wit)).sum *
                                          C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ))))))) +
                            -(X Vars_α *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                                  (to_MvPolynomial_Option Vars) (v_wit b))
                                              (List.finRange n_wit)).sum *
                                          C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ))))))) +
                          -(X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                              (to_MvPolynomial_Option Vars) (w_wit b))
                                          (List.finRange n_wit)).sum *
                                      C (prover.2 Proof_Idx.B SRS_Elements_Idx.δ))))))))))))) +
          (-(X Vars_γ *
                (X Vars_δ *
                  (X Vars_α *
                    (X Vars_γ *
                      (X Vars_δ *
                        (X Vars_γ *
                          (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) *
                            C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)))))))) +
            (-(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_β *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) *
                              C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)))))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_δ *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_γ *
                              (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) *
                                C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)))))))) +
                (-(X Vars_γ *
                      (X Vars_δ *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_γ *
                              ((List.map (fun b => C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum *
                                C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))))))) +
                  (-(X Vars_γ *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_γ *
                              ((List.map
                                    (fun b =>
                                      C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t b)) *
                                        (X Vars_x ^ ↑b *
                                          (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_β *
                              (X Vars_δ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (u_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))))))) +
                          -(X Vars_α *
                              (X Vars_δ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (v_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))))))) +
                        -(X Vars_δ *
                            (X Vars_γ *
                              (X Vars_δ *
                                (X Vars_γ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (w_stmt b))
                                        (List.finRange n_stmt)).sum *
                                    C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)))))) +
                      (-(X Vars_β *
                              (X Vars_γ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (u_wit b))
                                            (List.finRange n_wit)).sum *
                                        C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))))))) +
                          -(X Vars_α *
                              (X Vars_γ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (v_wit b))
                                            (List.finRange n_wit)).sum *
                                        C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))))))) +
                        -(X Vars_γ *
                            (X Vars_γ *
                              (X Vars_δ *
                                (X Vars_γ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (w_wit b))
                                        (List.finRange n_wit)).sum *
                                    C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))))))))))))) +
        (-(X Vars_γ *
              (X Vars_δ *
                (X Vars_α *
                  (X Vars_γ *
                    (X Vars_δ *
                      (X Vars_β *
                        (C (prover.1 Proof_Idx.A SRS_Elements_Idx.α) *
                          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_β *
                          (C (prover.1 Proof_Idx.A SRS_Elements_Idx.β) *
                            C (prover.2 Proof_Idx.B SRS_Elements_Idx.β)))))))) +
            (-(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_β *
                            (C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ) *
                              C (prover.2 Proof_Idx.B SRS_Elements_Idx.β)))))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_β *
                            ((List.map (fun b => C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                  (List.finRange n_var)).sum *
                              C (prover.2 Proof_Idx.B SRS_Elements_Idx.β))))))) +
                (-(X Vars_γ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_β *
                            ((List.map
                                  (fun b =>
                                    C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t b)) *
                                      (X Vars_x ^ ↑b *
                                        (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_β *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                              (to_MvPolynomial_Option Vars) (u_stmt b))
                                          (List.finRange n_stmt)).sum *
                                      C (prover.2 Proof_Idx.B SRS_Elements_Idx.β))))))) +
                        -(X Vars_α *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                              (to_MvPolynomial_Option Vars) (v_stmt b))
                                          (List.finRange n_stmt)).sum *
                                      C (prover.2 Proof_Idx.B SRS_Elements_Idx.β))))))) +
                      -(X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_β *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                          (to_MvPolynomial_Option Vars) (w_stmt b))
                                      (List.finRange n_stmt)).sum *
                                  C (prover.2 Proof_Idx.B SRS_Elements_Idx.β)))))) +
                    (-(X Vars_β *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                              (to_MvPolynomial_Option Vars) (u_wit b))
                                          (List.finRange n_wit)).sum *
                                      C (prover.2 Proof_Idx.B SRS_Elements_Idx.β))))))) +
                        -(X Vars_α *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                              (to_MvPolynomial_Option Vars) (v_wit b))
                                          (List.finRange n_wit)).sum *
                                      C (prover.2 Proof_Idx.B SRS_Elements_Idx.β))))))) +
                      -(X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_β *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) *
                                          (to_MvPolynomial_Option Vars) (w_wit b))
                                      (List.finRange n_wit)).sum *
                                  C (prover.2 Proof_Idx.B SRS_Elements_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 *
                        (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (u_stmt b))
                            (List.finRange n_stmt)).sum)))) +
              X Vars_α *
                (X Vars_δ *
                  (X Vars_γ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (v_stmt b))
                            (List.finRange n_stmt)).sum)))) +
            X Vars_δ *
              (X Vars_γ *
                (X Vars_δ *
                  (X Vars_γ *
                    (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (w_stmt b))
                        (List.finRange n_stmt)).sum))) +
          (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.α)))))) +
            (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.β)))))) +
              (X Vars_γ *
                  (X Vars_δ *
                    (X Vars_δ * (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_γ *
                        (X Vars_δ *
                          (X Vars_δ *
                            (List.map (fun b => C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                (List.finRange n_var)).sum)))) +
                  (X Vars_γ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_δ *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow_times_t b)) *
                                    (X Vars_x ^ ↑b *
                                      (to_MvPolynomial_Option Vars)
                                        (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                                (List.finRange (n_var - 1))).sum))) +
                    (X Vars_β *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (u_stmt b))
                                        (List.finRange n_stmt)).sum)))) +
                          X Vars_α *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (v_stmt b))
                                        (List.finRange n_stmt)).sum)))) +
                        X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y b)) *
                                        (to_MvPolynomial_Option Vars) (w_stmt b))
                                    (List.finRange n_stmt)).sum))) +
                      (X Vars_β *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (u_wit b))
                                        (List.finRange n_wit)).sum)))) +
                          X Vars_α *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (v_wit b))
                                        (List.finRange n_wit)).sum)))) +
                        X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q b)) *
                                        (to_MvPolynomial_Option Vars) (w_wit b))
                                    (List.finRange n_wit)).sum)))))))))))) =
    0
eqnA :
  X Vars_γ * (X Vars_δ * (X Vars_α * C (prover.1 Proof_Idx.A SRS_Elements_Idx.α))) +
      (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_δ * C (prover.1 Proof_Idx.A SRS_Elements_Idx.δ))) +
          (X Vars_γ *
              (X Vars_δ *
                (List.map (fun b => C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                    (List.finRange n_var)).sum) +
            (X Vars_γ *
                (List.map
                    (fun b =>
                      C (prover.1 Proof_Idx.A (SRS_Elements_Idx.x_pow_times_t b)) *
                        (X Vars_x ^ ↑b *
                          (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                    (List.finRange (n_var - 1))).sum +
              (X Vars_β *
                      (X Vars_δ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (u_stmt b))
                            (List.finRange n_stmt)).sum) +
                    X Vars_α *
                      (X Vars_δ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (v_stmt b))
                            (List.finRange n_stmt)).sum) +
                  X Vars_δ *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.y b)) * (to_MvPolynomial_Option Vars) (w_stmt b))
                        (List.finRange n_stmt)).sum +
                (X Vars_β *
                      (X Vars_γ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                            (List.finRange n_wit)).sum) +
                    X Vars_α *
                      (X Vars_γ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (v_wit b))
                            (List.finRange n_wit)).sum) +
                  X Vars_γ *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_Idx.A (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (w_wit b))
                        (List.finRange n_wit)).sum)))))) =
    X Vars_γ * (X Vars_δ * (X Vars_β * C (prover.2 Proof_Idx.A SRS_Elements_Idx.β))) +
      (X Vars_γ * (X Vars_δ * (X Vars_γ * C (prover.2 Proof_Idx.A SRS_Elements_Idx.γ))) +
        (X Vars_γ * (X Vars_δ * (X Vars_δ * C (prover.2 Proof_Idx.A SRS_Elements_Idx.δ))) +
          X Vars_γ *
            (X Vars_δ *
              (List.map (fun b => C (prover.2 Proof_Idx.A (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                  (List.finRange n_var)).sum)))
eqnB :
  X Vars_γ * (X Vars_δ * (X Vars_α * C (prover.1 Proof_Idx.B SRS_Elements_Idx.α))) +
      (X Vars_γ * (X Vars_δ * (X Vars_β * C (prover.1 Proof_Idx.B SRS_Elements_Idx.β))) +
        (X Vars_γ * (X Vars_δ * (X Vars_δ * C (prover.1 Proof_Idx.B SRS_Elements_Idx.δ))) +
          (X Vars_γ *
              (X Vars_δ *
                (List.map (fun b => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                    (List.finRange n_var)).sum) +
            (X Vars_γ *
                (List.map
                    (fun b =>
                      C (prover.1 Proof_Idx.B (SRS_Elements_Idx.x_pow_times_t b)) *
                        (X Vars_x ^ ↑b *
                          (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                    (List.finRange (n_var - 1))).sum +
              (X Vars_β *
                      (X Vars_δ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.B (SRS_Elements_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (u_stmt b))
                            (List.finRange n_stmt)).sum) +
                    X Vars_α *
                      (X Vars_δ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.B (SRS_Elements_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (v_stmt b))
                            (List.finRange n_stmt)).sum) +
                  X Vars_δ *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_Idx.B (SRS_Elements_Idx.y b)) * (to_MvPolynomial_Option Vars) (w_stmt b))
                        (List.finRange n_stmt)).sum +
                (X Vars_β *
                      (X Vars_γ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.B (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                            (List.finRange n_wit)).sum) +
                    X Vars_α *
                      (X Vars_γ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.B (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (v_wit b))
                            (List.finRange n_wit)).sum) +
                  X Vars_γ *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_Idx.B (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (w_wit b))
                        (List.finRange n_wit)).sum)))))) =
    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_γ * C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ))) +
        (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 *
              (List.map (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                  (List.finRange n_var)).sum)))
eqnC :
  X Vars_γ * (X Vars_δ * (X Vars_α * C (prover.1 Proof_Idx.C SRS_Elements_Idx.α))) +
      (X Vars_γ * (X Vars_δ * (X Vars_β * C (prover.1 Proof_Idx.C SRS_Elements_Idx.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 *
                (List.map (fun b => C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                    (List.finRange n_var)).sum) +
            (X Vars_γ *
                (List.map
                    (fun b =>
                      C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow_times_t b)) *
                        (X Vars_x ^ ↑b *
                          (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                    (List.finRange (n_var - 1))).sum +
              (X Vars_β *
                      (X Vars_δ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (u_stmt b))
                            (List.finRange n_stmt)).sum) +
                    X Vars_α *
                      (X Vars_δ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (v_stmt b))
                            (List.finRange n_stmt)).sum) +
                  X Vars_δ *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.y b)) * (to_MvPolynomial_Option Vars) (w_stmt b))
                        (List.finRange n_stmt)).sum +
                (X Vars_β *
                      (X Vars_γ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                            (List.finRange n_wit)).sum) +
                    X Vars_α *
                      (X Vars_γ *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (v_wit b))
                            (List.finRange n_wit)).sum) +
                  X Vars_γ *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q b)) * (to_MvPolynomial_Option Vars) (w_wit b))
                        (List.finRange n_wit)).sum)))))) =
    X Vars_γ * (X Vars_δ * (X Vars_β * C (prover.2 Proof_Idx.C SRS_Elements_Idx.β))) +
      (X Vars_γ * (X Vars_δ * (X Vars_γ * C (prover.2 Proof_Idx.C SRS_Elements_Idx.γ))) +
        (X Vars_γ * (X Vars_δ * (X Vars_δ * C (prover.2 Proof_Idx.C SRS_Elements_Idx.δ))) +
          X Vars_γ *
            (X Vars_δ *
              (List.map (fun b => C (prover.2 Proof_Idx.C (SRS_Elements_Idx.x_pow b)) * X Vars_x ^ ↑b)
                  (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