Theorem: BabySNARK.is_sound
Proof state before:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnH :
  (List.map (fun x => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum))
eqnV :
  (List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum))
eqnB :
  (List.map (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum))
eqnI :
  (List.map (fun x => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
          (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) * X Vars_β *
              (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
            (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) *
                (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
              (List.map
                    (fun x =>
                      C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) *
                        (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                    (List.finRange n_wit)).sum *
                (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum)) +
        (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
          (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      ((List.map (fun x => C (if ↑x = 0 then 1 else 0) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
          (List.map (fun x => C (if ↑x = 0 then 1 else 0) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
        ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                          (List.finRange n_var)).sum *
                      -(List.map
                            (fun x =>
                              C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                            (List.finRange n_wit)).sum +
                    (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β *
                        -(List.map
                              (fun x =>
                                C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                  (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                              (List.finRange n_wit)).sum +
                      (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) *
                          -(List.map
                                (fun x =>
                                  C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                    (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                                (List.finRange n_wit)).sum +
                        (List.map
                              (fun x =>
                                C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                  (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                              (List.finRange n_wit)).sum *
                          -(List.map
                                (fun x =>
                                  C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                    (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                                (List.finRange n_wit)).sum)) +
                  (List.map
                        (fun x =>
                          C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            X Vars_τ ^ ↑x)
                        (List.finRange n_var)).sum *
                    -(List.map
                          (fun x =>
                            C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                              (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                          (List.finRange n_wit)).sum +
                ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                          (List.finRange n_var)).sum *
                      -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)) +
                    (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β *
                        -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)) +
                      (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) *
                          -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)) +
                        (List.map
                              (fun x =>
                                C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                  (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)))) +
                  (List.map
                        (fun x =>
                          C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            X Vars_τ ^ ↑x)
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β))) +
              ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β) +
                  (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β *
                      -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β) +
                    (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) *
                        -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β) +
                      (List.map
                            (fun x =>
                              C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                                (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                            (List.finRange n_wit)).sum *
                        -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β))) +
                (List.map
                      (fun x =>
                        C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x)
                      (List.finRange n_var)).sum *
                  -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β)) +
            ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                      (List.finRange n_var)).sum *
                  -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                        (List.finRange n_var)).sum +
                (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β *
                    -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                          (List.finRange n_var)).sum +
                  (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) *
                      -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                            (List.finRange n_var)).sum +
                    (List.map
                          (fun x =>
                            C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                              (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                          (List.finRange n_wit)).sum *
                      -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                            (List.finRange n_var)).sum)) +
              (List.map
                    (fun x =>
                      C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x)
                    (List.finRange n_var)).sum *
                -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                      (List.finRange n_var)).sum) +
          ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          X Vars_τ ^ ↑x))
                    (List.finRange n_var)).sum +
              (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β *
                  (List.map
                      (fun x =>
                        -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            X Vars_τ ^ ↑x))
                      (List.finRange n_var)).sum +
                (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) *
                    (List.map
                        (fun x =>
                          -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              X Vars_τ ^ ↑x))
                        (List.finRange n_var)).sum +
                  (List.map
                        (fun x =>
                          C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                            (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                        (List.finRange n_wit)).sum *
                    (List.map
                        (fun x =>
                          -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              X Vars_τ ^ ↑x))
                        (List.finRange n_var)).sum)) +
            (List.map
                  (fun x =>
                    C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map
                  (fun x =>
                    -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x))
                  (List.finRange n_var)).sum))) =
    0
eqnII :
  (List.map (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
          X Vars_β +
        (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ) * X Vars_β * X Vars_β +
          (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) * X Vars_β +
            (List.map
                  (fun x =>
                    C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) *
                      (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                  (List.finRange n_wit)).sum *
              X Vars_β)) +
      (X Vars_γ * X Vars_β *
              -(List.map
                    (fun x =>
                      C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                        (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                    (List.finRange n_wit)).sum +
            X Vars_γ * X Vars_β * -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)) +
          X Vars_γ * X Vars_β * -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β) +
        X Vars_γ * X Vars_β *
          -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                (List.finRange n_var)).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_Idx.B (SRS_Elements_Idx.βu i)) * u_wit i)
              (List.finRange n_wit)).sum) ^
        2 -
      1 =
    t *
      (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu i)) * u_wit i)
          (List.finRange n_wit)).sum
Tactic:
simp only [
  mul_assoc,
  mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,
  mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),
  neg_mul, mul_neg,
  List.sum_map_mul_right, List.sum_map_mul_left] at eqnI eqnII eqnH eqnV eqnB
Proof state after:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnI :
  (List.map (fun x => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
          (X Vars_β *
              (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum) +
            (X Vars_γ *
                (X Vars_β *
                  (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum)) +
              X Vars_β *
                ((List.map
                      (fun b =>
                        C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum))) +
        (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
          (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      ((List.map (fun x => C (if ↑x = 0 then 1 else 0) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
          (List.map (fun x => C (if ↑x = 0 then 1 else 0) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
        (-(X Vars_β *
                        ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map
                              (fun b =>
                                C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                  (to_MvPolynomial_Option Vars) (u_wit b))
                              (List.finRange n_wit)).sum)) +
                    (-(X Vars_β *
                          (X Vars_β *
                            (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map
                                  (fun b =>
                                    C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                      (to_MvPolynomial_Option Vars) (u_wit b))
                                  (List.finRange n_wit)).sum))) +
                      (-(X Vars_γ *
                            (X Vars_β *
                              (X Vars_β *
                                (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                  (List.map
                                      (fun b =>
                                        C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                          (to_MvPolynomial_Option Vars) (u_wit b))
                                      (List.finRange n_wit)).sum)))) +
                        -(X Vars_β *
                            (X Vars_β *
                              ((List.map
                                    (fun b =>
                                      C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                        (to_MvPolynomial_Option Vars) (u_wit b))
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun b =>
                                      C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                        (to_MvPolynomial_Option Vars) (u_wit b))
                                    (List.finRange n_wit)).sum))))) +
                  -(X Vars_β *
                      ((List.map
                            (fun x =>
                              C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                X Vars_τ ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun b =>
                              C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                (to_MvPolynomial_Option Vars) (u_wit b))
                            (List.finRange n_wit)).sum)) +
                (-(X Vars_γ *
                        (X Vars_β *
                          ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                                (List.finRange n_var)).sum *
                            C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    (-(X Vars_β *
                          (X Vars_γ *
                            (X Vars_β *
                              (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                                C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))))) +
                      (-(X Vars_γ *
                            (X Vars_β *
                              (X Vars_γ *
                                (X Vars_β *
                                  (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                    C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))))) +
                        -(X Vars_β *
                            (X Vars_γ *
                              (X Vars_β *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                          (to_MvPolynomial_Option Vars) (u_wit b))
                                      (List.finRange n_wit)).sum *
                                  C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))))))) +
                  -(X Vars_γ *
                      (X Vars_β *
                        ((List.map
                              (fun x =>
                                C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  X Vars_τ ^ ↑x)
                              (List.finRange n_var)).sum *
                          C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))))) +
              (-(X Vars_β *
                      ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                            (List.finRange n_var)).sum *
                        C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
                  (-(X Vars_β *
                        (X Vars_β *
                          (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                    (-(X Vars_γ *
                          (X Vars_β *
                            (X Vars_β *
                              (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))))) +
                      -(X Vars_β *
                          (X Vars_β *
                            ((List.map
                                  (fun b =>
                                    C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                      (to_MvPolynomial_Option Vars) (u_wit b))
                                  (List.finRange n_wit)).sum *
                              C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))))) +
                -(X Vars_β *
                    ((List.map
                          (fun x =>
                            C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              X Vars_τ ^ ↑x)
                          (List.finRange n_var)).sum *
                      C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
            (-((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                        (List.finRange n_var)).sum) +
                (-(X Vars_β *
                      (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(X Vars_γ *
                        (X Vars_β *
                          (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                                (List.finRange n_var)).sum))) +
                    -(X Vars_β *
                        ((List.map
                              (fun b =>
                                C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu b)) *
                                  (to_MvPolynomial_Option Vars) (u_wit b))
                              (List.finRange n_wit)).sum *
                          (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                              (List.finRange n_var)).sum)))) +
              -((List.map
                      (fun x =>
                        C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                      (List.finRange n_var)).sum)) +
          ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          X Vars_τ ^ ↑x))
                    (List.finRange n_var)).sum +
              (X Vars_β *
                  (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              X Vars_τ ^ ↑x))
                        (List.finRange n_var)).sum) +
                (X Vars_γ *
                    (X Vars_β *
                      (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  X Vars_τ ^ ↑x))
                            (List.finRange n_var)).sum)) +
                  X Vars_β *
                    ((List.map
                          (fun b =>
                            C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                X Vars_τ ^ ↑x))
                          (List.finRange n_var)).sum))) +
            (List.map
                  (fun x =>
                    C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map
                  (fun x =>
                    -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x))
                  (List.finRange n_var)).sum))) =
    0
eqnII :
  X Vars_β *
          (List.map (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
              (List.finRange n_var)).sum +
        (X Vars_β * (X Vars_β * C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          (X Vars_γ * (X Vars_β * (X Vars_β * C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ))) +
            X Vars_β *
              (X Vars_β *
                (List.map
                    (fun b =>
                      C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                    (List.finRange n_wit)).sum))) +
      (-(X Vars_γ *
                (X Vars_β *
                  (X Vars_β *
                    (List.map
                        (fun b =>
                          C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                        (List.finRange n_wit)).sum))) +
            -(X Vars_γ * (X Vars_β * (X Vars_γ * (X Vars_β * C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))))) +
          -(X Vars_γ * (X Vars_β * (X Vars_β * C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
        -(X Vars_γ *
            (X Vars_β *
              (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                  (List.finRange n_var)).sum))) =
    0
eqnH :
  (List.map (fun x => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (X Vars_β * C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) +
        (X Vars_γ * (X Vars_β * C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
          X Vars_β *
            (List.map
                (fun b => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (X Vars_β * C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ) +
        (X Vars_γ * (X Vars_β * C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
          X Vars_β *
            (List.map
                (fun b => C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                (List.finRange n_wit)).sum))
eqnV :
  (List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (X Vars_β * C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) +
        (X Vars_γ * (X Vars_β * C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
          X Vars_β *
            (List.map
                (fun b => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (X Vars_β * C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) +
        (X Vars_γ * (X Vars_β * C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
          X Vars_β *
            (List.map
                (fun b => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                (List.finRange n_wit)).sum))
eqnB :
  (List.map (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (X Vars_β * C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ) +
        (X Vars_γ * (X Vars_β * C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
          X Vars_β *
            (List.map
                (fun b => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (X Vars_β * C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) +
        (X Vars_γ * (X Vars_β * C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
          X Vars_β *
            (List.map
                (fun b => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu b)) * (to_MvPolynomial_Option Vars) (u_wit b))
                (List.finRange n_wit)).sum))
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
          (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu i)) * u_wit i)
              (List.finRange n_wit)).sum) ^
        2 -
      1 =
    t *
      (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu i)) * u_wit i)
          (List.finRange n_wit)).sum