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 :
  (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * 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.y) ^ 5 * X none ^ ↑x *
                            (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                            (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (v_stmt x) +
                              X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt x)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                            (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (u_wit x) +
                                X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (v_wit x) +
                              X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit x)))
                        (List.finRange n_wit)).sum)))))) *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25))) +
      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75 *
          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
        ((List.map
                (fun x =>
                  C (stmt x) *
                    (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt x) +
                        X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (v_stmt x) +
                      X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt x)))
                (List.finRange n_stmt)).sum *
            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * 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.y) ^ 5 * X none ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                    X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (v_stmt x) +
                                  X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt)).sum +
                        (List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (u_wit x) +
                                    X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (v_wit x) +
                                  X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit x)))
                            (List.finRange n_wit)).sum)))))) *
            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1))) =
    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 [mul_add, add_mul, List.sum_map_add] at eqn
Proof state after:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
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 :
  C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) *
                -(List.map
                      (fun x =>
                        C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                      (List.finRange n_var)).sum +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) *
                  -(List.map
                        (fun x =>
                          C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                        (List.finRange n_var)).sum +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) *
                    -(List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * 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 x)) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(List.map
                            (fun x =>
                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * 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.y) ^ 5 * X none ^ ↑x *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                            (List.finRange (n_var - 1))).sum *
                        -(List.map
                              (fun x =>
                                C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                              (List.finRange n_var)).sum +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 *
                                        (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                        (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                                  (List.finRange n_var)).sum +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 *
                                        (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                        (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.y) ^ 5 * (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.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                                  (List.finRange n_var)).sum)))))) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X (some Vars.y) ^ 5 * X none ^ ↑x *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                            (List.finRange (n_var - 1))).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 *
                                        (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                        (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 *
                                        (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                        (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1))))))))) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X (some Vars.y) ^ 5 * X none ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5))))))))) +
        (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) *
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
              ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                      (List.finRange n_var)).sum *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                            (X (some Vars.y) ^ 5 * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1))).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                  ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 *
                                    (to_MvPolynomial_Option Vars) (u_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                    (to_MvPolynomial_Option Vars) (v_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)))
                            (List.finRange n_stmt)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                    ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 *
                                    (to_MvPolynomial_Option Vars) (u_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                    (to_MvPolynomial_Option Vars) (v_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit i)))
                            (List.finRange n_wit)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25))))))))) +
      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75 *
          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
        ((List.map
                    (fun i =>
                      C (stmt i) *
                        (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5) +
              (List.map
                    (fun i =>
                      C (stmt i) *
                        (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (v_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5) +
            (List.map (fun i => C (stmt i) * (X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)))
                  (List.finRange n_stmt)).sum *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
            (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                        (List.finRange n_var)).sum *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X (some Vars.y) ^ 5 * X none ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum *
                      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                  (X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                  (X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)))))))))) =
    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