Theorem: Groth16TypeIII.is_sound
Proof state before:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
r : Fin n_wit → F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  [[C 1 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C (-1) *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C (-1) *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
          ([C 1 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum),
        [C 0 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
            ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
          ([C 0 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum),
        [C 0 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C (stmt x) * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C (stmt x) *
                              (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C (stmt x) * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
          ([C 0 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C (stmt x) * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C (stmt x) *
                              (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C (stmt x) * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum),
        [C 0 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 1 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                          [C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var)).sum].sum +
          ([C 0 *
                    ([C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 1 *
                    ([C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                                  [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                                      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                                [C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                              List.map
                                (fun x =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                    (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                                (List.finRange n_var) ++
                            List.map
                              (fun x =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                  (X (some Vars.γ) * X none ^ ↑x *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum)].sum =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x =>
            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum
Tactic:
simp only [map_one, List.singleton_append, List.cons_append, map_prod, map_sub, List.append_assoc,
  List.sum_cons, List.sum_append, List.sum_map_add, one_mul, map_zero, zero_mul, List.sum_nil,
  add_zero, map_neg, neg_mul, neg_add_rev, List.map_const', List.length_finRange,
  List.sum_replicate, smul_zero, mul_zero, zero_add] at eqn
Proof state after:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
r : Fin n_wit → F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α)) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                        (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
                ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                          (X (some Vars.γ) * X none ^ ↑x *
                            ∏ x : Fin n_wit,
                              ((to_MvPolynomial_Option Vars) Polynomial.X -
                                (to_MvPolynomial_Option Vars) (Polynomial.C (r x)))))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i)))
                            (List.finRange n_stmt)).sum +
                        (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i)))
                            (List.finRange n_stmt)).sum +
                      (List.map
                          (fun i =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                              (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)))
                          (List.finRange n_stmt)).sum +
                    ((List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i)))
                            (List.finRange n_wit)).sum +
                        (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i)))
                            (List.finRange n_wit)).sum +
                      (List.map
                          (fun i =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                              (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit i)))
                          (List.finRange n_wit)).sum))))))) *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                        (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                  (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))) +
          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)))) +
      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)) +
        (((List.map
                    (fun i =>
                      C (stmt i) * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i)))
                    (List.finRange n_stmt)).sum +
                (List.map
                    (fun i =>
                      C (stmt i) * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i)))
                    (List.finRange n_stmt)).sum +
              (List.map (fun i => C (stmt i) * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)))
                  (List.finRange n_stmt)).sum) *
            (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ)) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α)) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                  (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                        (List.finRange n_var)).sum +
                    ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X (some Vars.γ) * X none ^ ↑x *
                                ∏ x : Fin n_wit,
                                  ((to_MvPolynomial_Option Vars) Polynomial.X -
                                    (to_MvPolynomial_Option Vars) (Polynomial.C (r x)))))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum +
                            (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum +
                          (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum +
                        ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum +
                            (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum +
                          (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                  (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum))))))) *
            (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x =>
            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum