Theorem: Lipmaa.soundness
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 : Lipmaa.Stmt
prover : Prover F Lipmaa
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  -((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)) *
                            (List.map
                                (fun x =>
                                  (optionEquivRight F Vars)
                                      (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                    (optionEquivRight F Vars) (X none) ^ ↑x)
                                (List.finRange n_var)).sum)))))) +
              (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)) *
                              (List.map
                                  (fun x =>
                                    (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                      (optionEquivRight F Vars) (X none) ^ ↑x)
                                  (List.finRange n_var)).sum)))))) +
                (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)) *
                                (List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                        (optionEquivRight F Vars) (X none) ^ ↑x)
                                    (List.finRange n_var)).sum)))))) +
                  (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x))) *
                                        (optionEquivRight F Vars) (X none) ^ ↑x)
                                    (List.finRange n_var)).sum *
                                (List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                        (optionEquivRight F Vars) (X none) ^ ↑x)
                                    (List.finRange n_var)).sum))))) +
                    (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                        ((optionEquivRight F Vars) (X none) ^ ↑x *
                                          (optionEquivRight F Vars)
                                            ((to_MvPolynomial_Option Vars)
                                              (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))))
                                    (List.finRange (n_var - 1))).sum *
                                (List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                        (optionEquivRight F Vars) (X none) ^ ↑x)
                                    (List.finRange n_var)).sum)))) +
                      (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_stmt x)))
                                            (List.finRange n_stmt)).sum *
                                        (List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                                (optionEquivRight F Vars) (X none) ^ ↑x)
                                            (List.finRange n_var)).sum))))) +
                            -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_stmt x)))
                                            (List.finRange n_stmt)).sum *
                                        (List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                                (optionEquivRight F Vars) (X none) ^ ↑x)
                                            (List.finRange n_var)).sum))))) +
                          -((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_stmt x)))
                                        (List.finRange n_stmt)).sum *
                                    (List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                            (optionEquivRight F Vars) (X none) ^ ↑x)
                                        (List.finRange n_var)).sum)))) +
                        (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_wit x)))
                                            (List.finRange n_wit)).sum *
                                        (List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                                (optionEquivRight F Vars) (X none) ^ ↑x)
                                            (List.finRange n_var)).sum))))) +
                            -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_wit x)))
                                            (List.finRange n_wit)).sum *
                                        (List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                                (optionEquivRight F Vars) (X none) ^ ↑x)
                                            (List.finRange n_var)).sum))))) +
                          -((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_wit x)))
                                        (List.finRange n_wit)).sum *
                                    (List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                            (optionEquivRight F Vars) (X none) ^ ↑x)
                                        (List.finRange n_var)).sum)))))))))) +
            (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)) *
                              (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))))) +
              (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)) *
                                (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))))) +
                (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)) *
                                  (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))))) +
                  (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((List.map
                                      (fun x =>
                                        (optionEquivRight F Vars)
                                            (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x))) *
                                          (optionEquivRight F Vars) (X none) ^ ↑x)
                                      (List.finRange n_var)).sum *
                                  (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                    (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((List.map
                                      (fun x =>
                                        (optionEquivRight F Vars)
                                            (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                          ((optionEquivRight F Vars) (X none) ^ ↑x *
                                            (optionEquivRight F Vars)
                                              ((to_MvPolynomial_Option Vars)
                                                (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))))
                                      (List.finRange (n_var - 1))).sum *
                                  (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                      (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                (optionEquivRight F Vars)
                                                    (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                                  (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_stmt x)))
                                              (List.finRange n_stmt)).sum *
                                          (optionEquivRight F Vars)
                                            (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                            -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                (optionEquivRight F Vars)
                                                    (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                                  (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_stmt x)))
                                              (List.finRange n_stmt)).sum *
                                          (optionEquivRight F Vars)
                                            (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                          -((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    ((List.map
                                          (fun x =>
                                            (optionEquivRight F Vars)
                                                (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                              (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_stmt x)))
                                          (List.finRange n_stmt)).sum *
                                      (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                        (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                (optionEquivRight F Vars)
                                                    (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                                  (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_wit x)))
                                              (List.finRange n_wit)).sum *
                                          (optionEquivRight F Vars)
                                            (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                            -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                        ((List.map
                                              (fun x =>
                                                (optionEquivRight F Vars)
                                                    (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                                  (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_wit x)))
                                              (List.finRange n_wit)).sum *
                                          (optionEquivRight F Vars)
                                            (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                          -((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    ((List.map
                                          (fun x =>
                                            (optionEquivRight F Vars)
                                                (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                              (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_wit x)))
                                          (List.finRange n_wit)).sum *
                                      (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))))))))) +
          (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)) *
                            (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))))) +
            (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)) *
                              (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))))) +
              (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)) *
                                (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))))) +
                (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x))) *
                                        (optionEquivRight F Vars) (X none) ^ ↑x)
                                    (List.finRange n_var)).sum *
                                (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                  (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                        ((optionEquivRight F Vars) (X none) ^ ↑x *
                                          (optionEquivRight F Vars)
                                            ((to_MvPolynomial_Option Vars)
                                              (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))))
                                    (List.finRange (n_var - 1))).sum *
                                (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                    (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_stmt x)))
                                            (List.finRange n_stmt)).sum *
                                        (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                          -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_stmt x)))
                                            (List.finRange n_stmt)).sum *
                                        (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                        -((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_stmt x)))
                                        (List.finRange n_stmt)).sum *
                                    (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                      (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_wit x)))
                                            (List.finRange n_wit)).sum *
                                        (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                          -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                      ((List.map
                                            (fun x =>
                                              (optionEquivRight F Vars)
                                                  (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_wit x)))
                                            (List.finRange n_wit)).sum *
                                        (optionEquivRight F Vars)
                                          (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                        -((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                  ((List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_wit x)))
                                        (List.finRange n_wit)).sum *
                                    (optionEquivRight F Vars)
                                      (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))))))))) +
        (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                        ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)) *
                          (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))))) +
          (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                          ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)) *
                            (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))))) +
            (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)) *
                              (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))))) +
              (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((List.map
                                  (fun x =>
                                    (optionEquivRight F Vars)
                                        (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x))) *
                                      (optionEquivRight F Vars) (X none) ^ ↑x)
                                  (List.finRange n_var)).sum *
                              (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
                (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((List.map
                                  (fun x =>
                                    (optionEquivRight F Vars)
                                        (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                      ((optionEquivRight F Vars) (X none) ^ ↑x *
                                        (optionEquivRight F Vars)
                                          ((to_MvPolynomial_Option Vars)
                                            (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))))
                                  (List.finRange (n_var - 1))).sum *
                              (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                  (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                    ((List.map
                                          (fun x =>
                                            (optionEquivRight F Vars)
                                                (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                              (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_stmt x)))
                                          (List.finRange n_stmt)).sum *
                                      (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
                        -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                    ((List.map
                                          (fun x =>
                                            (optionEquivRight F Vars)
                                                (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                              (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_stmt x)))
                                          (List.finRange n_stmt)).sum *
                                      (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
                      -((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                ((List.map
                                      (fun x =>
                                        (optionEquivRight F Vars)
                                            (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) *
                                          (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_stmt x)))
                                      (List.finRange n_stmt)).sum *
                                  (optionEquivRight F Vars) (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                    (-((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                    ((List.map
                                          (fun x =>
                                            (optionEquivRight F Vars)
                                                (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                              (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_wit x)))
                                          (List.finRange n_wit)).sum *
                                      (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
                        -((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                    ((List.map
                                          (fun x =>
                                            (optionEquivRight F Vars)
                                                (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                              (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_wit x)))
                                          (List.finRange n_wit)).sum *
                                      (optionEquivRight F Vars)
                                        (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
                      -((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                                ((List.map
                                      (fun x =>
                                        (optionEquivRight F Vars)
                                            (C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
                                          (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_wit x)))
                                      (List.finRange n_wit)).sum *
                                  (optionEquivRight F Vars)
                                    (C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))))))))) +
      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  (optionEquivRight F Vars) (X (some Vars.y)) ^ 25)))) +
        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        (List.map
                            (fun x =>
                              (optionEquivRight F Vars) (C (stmt x)) *
                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_stmt x)))
                            (List.finRange n_stmt)).sum)))) +
              (optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        (List.map
                            (fun x =>
                              (optionEquivRight F Vars) (C (stmt x)) *
                                (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_stmt x)))
                            (List.finRange n_stmt)).sum)))) +
            (optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    (List.map
                        (fun x =>
                          (optionEquivRight F Vars) (C (stmt x)) *
                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_stmt x)))
                        (List.finRange n_stmt)).sum))) +
          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        (optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α))))))) +
            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          (optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β))))))) +
              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            (optionEquivRight F Vars) (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ))))))) +
                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            (List.map
                                (fun x =>
                                  (optionEquivRight F Vars)
                                      (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x))) *
                                    (optionEquivRight F Vars) (X none) ^ ↑x)
                                (List.finRange n_var)).sum)))) +
                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                        ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                            (List.map
                                (fun x =>
                                  (optionEquivRight F Vars)
                                      (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                    ((optionEquivRight F Vars) (X none) ^ ↑x *
                                      (optionEquivRight F Vars)
                                        ((to_MvPolynomial_Option Vars)
                                          (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))))
                                (List.finRange (n_var - 1))).sum))) +
                    ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    (List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_stmt x)))
                                        (List.finRange n_stmt)).sum)))) +
                          (optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    (List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_stmt x)))
                                        (List.finRange n_stmt)).sum)))) +
                        (optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                (List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                        (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_stmt x)))
                                    (List.finRange n_stmt)).sum))) +
                      ((optionEquivRight F Vars) (X (some Vars.y)) ^ 25 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    (List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (u_wit x)))
                                        (List.finRange n_wit)).sum)))) +
                          (optionEquivRight F Vars) (X (some Vars.y)) ^ 75 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                                ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                  ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                    (List.map
                                        (fun x =>
                                          (optionEquivRight F Vars)
                                              (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                            (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (v_wit x)))
                                        (List.finRange n_wit)).sum)))) +
                        (optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                          ((optionEquivRight F Vars) (X (some Vars.y)) ^ 5 *
                            ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                              ((optionEquivRight F Vars) (X (some Vars.y)) ^ 1 *
                                (List.map
                                    (fun x =>
                                      (optionEquivRight F Vars)
                                          (C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                        (optionEquivRight F Vars) ((to_MvPolynomial_Option Vars) (w_wit x)))
                                    (List.finRange n_wit)).sum)))))))))))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (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 [optionEquivRight_C, optionEquivRight_X_none, optionEquivRight_X_some, optionEquivRight_to_MvPolynomial_Option] 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 : Lipmaa.Stmt
prover : Prover F Lipmaa
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  -(X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 75 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)) *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                                    C Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum)))))) +
              (-(X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 25 *
                        (X Vars.y ^ 5 *
                          (X Vars.y ^ 1 *
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                                            (fun x =>
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                                        (fun x =>
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                                            (fun x =>
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                                            (fun x =>
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                                        (fun x =>
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                        (X Vars.y ^ 1 *
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                                            (Polynomial.C
                                              (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
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                                              (fun x =>
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                                              (fun x =>
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                                              (fun x =>
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                                              (List.finRange n_wit)).sum *
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                          -(X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
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                                          (List.finRange n_wit)).sum *
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                                    (fun x =>
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                                        C Polynomial.X ^ ↑x)
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                        (X Vars.y ^ 5 *
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                                            (fun x =>
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                                            (List.finRange n_stmt)).sum *
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                                            (fun x =>
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                                            (List.finRange n_stmt)).sum *
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                                        (fun x =>
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                                (X Vars.y ^ 5 *
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                                            (fun x =>
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                                            (List.finRange n_wit)).sum *
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                                (X Vars.y ^ 5 *
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                                            (fun x =>
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                                            (List.finRange n_wit)).sum *
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                              (X Vars.y ^ 1 *
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                                        (fun x =>
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                                        (List.finRange n_wit)).sum *
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                                  (fun x =>
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                                      C Polynomial.X ^ ↑x)
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                (-(X Vars.y ^ 5 *
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                                  (fun x =>
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                                          (fun x =>
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                                          (fun x =>
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                                          (fun x =>
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                                          (fun x =>
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                                          (List.finRange n_wit)).sum *
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                                        C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) *
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                                      (List.finRange n_wit)).sum *
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                  (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (List.map (fun x => C (Polynomial.C (stmt x)) * C (v_stmt x)) (List.finRange n_stmt)).sum)))) +
            X Vars.y ^ 1 *
              (X Vars.y ^ 5 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 5 *
                    (List.map (fun x => C (Polynomial.C (stmt x)) * C (w_stmt x)) (List.finRange n_stmt)).sum))) +
          (X Vars.y ^ 5 *
              (X Vars.y ^ 1 *
                (X Vars.y ^ 75 *
                  (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 1 * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α))))))) +
            (X Vars.y ^ 5 *
                (X Vars.y ^ 1 *
                  (X Vars.y ^ 25 *
                    (X Vars.y ^ 5 *
                      (X Vars.y ^ 1 *
                        (X Vars.y ^ 1 * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β))))))) +
              (X Vars.y ^ 5 *
                  (X Vars.y ^ 1 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ))))))) +
                (X Vars.y ^ 5 *
                    (X Vars.y ^ 1 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x))) *
                                    C Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum)))) +
                  (X Vars.y ^ 5 *
                      (X Vars.y ^ 5 *
                        (X Vars.y ^ 1 *
                          (X Vars.y ^ 1 *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                    (C Polynomial.X ^ ↑x * C (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                                (List.finRange (n_var - 1))).sum))) +
                    (X Vars.y ^ 25 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    (List.map
                                        (fun x =>
                                          C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                            C (u_stmt x))
                                        (List.finRange n_stmt)).sum)))) +
                          X Vars.y ^ 75 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    (List.map
                                        (fun x =>
                                          C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                            C (v_stmt x))
                                        (List.finRange n_stmt)).sum)))) +
                        X Vars.y ^ 1 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                (List.map
                                    (fun x =>
                                      C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                        C (w_stmt x))
                                    (List.finRange n_stmt)).sum))) +
                      (X Vars.y ^ 25 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    (List.map
                                        (fun x =>
                                          C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                            C (u_wit x))
                                        (List.finRange n_wit)).sum)))) +
                          X Vars.y ^ 75 *
                            (X Vars.y ^ 5 *
                              (X Vars.y ^ 5 *
                                (X Vars.y ^ 1 *
                                  (X Vars.y ^ 1 *
                                    (List.map
                                        (fun x =>
                                          C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                            C (v_wit x))
                                        (List.finRange n_wit)).sum)))) +
                        X Vars.y ^ 5 *
                          (X Vars.y ^ 5 *
                            (X Vars.y ^ 1 *
                              (X Vars.y ^ 1 *
                                (List.map
                                    (fun x =>
                                      C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                        C (w_wit x))
                                    (List.finRange n_wit)).sum)))))))))))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (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