Theorem: BabySNARK.is_sound (FormalSnarksProject/SNARKs/BabySnark.lean)

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
hI_00 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0) 0
hI_01 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1) 0
hI_02 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2) 0
hI_10 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0) 0
hI_11 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1) 0
hI_12 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2) 0
hI_20 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0) 0
hI_21 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1) 0
hI_22 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
              ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
            ((monomial fun₀ | Vars.β => 1)
                (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
              ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                  (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
                (monomial fun₀ | Vars.β => 1)
                  ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                        (List.finRange n_wit)).sum *
                    (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum))) +
          (monomial 0)
            ((List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum) +
        ((monomial 0)
            ((List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum) +
          (-(monomial fun₀ | Vars.β => 1)
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum) +
                      (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum) +
                          -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                (List.map
                                    (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum))) +
                    -(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum) +
                  (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          ((List.map
                                (fun x =>
                                  Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                      (-(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                        (-(monomial
                                ((fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
                          -(monomial ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)))
                              ((List.map
                                    (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                    (List.finRange n_wit)).sum *
                                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))) +
                    -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                        ((List.map
                              (fun x =>
                                Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
                (-(monomial fun₀ | Vars.β => 1)
                        ((List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                    (-(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                      (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
                        -(monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                            ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                  -(monomial fun₀ | Vars.β => 1)
                      ((List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))) +
              ((monomial 0)
                    (-((List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum)) +
                  (-(monomial fun₀ | Vars.β => 1)
                        (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (-(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                          (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum) +
                      -(monomial fun₀ | Vars.β => 1)
                          ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            (List.map
                                (fun x =>
                                  Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                                (List.finRange n_var)).sum))) +
                (monomial 0)
                  (-((List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum))) +
            ((monomial 0)
                  ((List.map
                        (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum) +
                ((monomial fun₀ | Vars.β => 1)
                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum) +
                  ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
                      (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum) +
                    (monomial fun₀ | Vars.β => 1)
                      ((List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum *
                        (List.map
                            (fun x =>
                              -(Polynomial.C
                                    (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                  Polynomial.X ^ ↑x))
                            (List.finRange n_var)).sum))) +
              (monomial 0)
                ((List.map
                      (fun x =>
                        Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum))))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2) 0
hII_00 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0) 0
hII_01 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1) 0
hII_02 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2) 0
hII_10 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0) 0
hII_11 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1) 0
hII_12 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2) 0
hII_20 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0) 0
hII_21 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1) 0
hII_22 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial fun₀ | Vars.β => 1)
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum +
          ((monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
            ((monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
              (monomial ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1))
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum)) +
        (-(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                  (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum +
              -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))))
                  (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            -(monomial ((fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1)))
                (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          -(monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum)) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2) 0
hH_00 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_01 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_02 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_10 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_11 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_12 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_20 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_21 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hH_22 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_00 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_01 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_02 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_10 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_11 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_12 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_20 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_21 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hV_22 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_00 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_01 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_02 :
  coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_10 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_11 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_12 :
  coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_20 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_21 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)))
hB_22 :
  coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))) =
    coeff ((fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2)
      ((monomial 0)
          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum +
        ((monomial fun₀ | Vars.β => 1) (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
          ((monomial ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1))
              (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)) +
            (monomial fun₀ | Vars.β => 1)
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (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
Tactic:
simp only [coeff_monomial, coeff_add, coeff_neg, coeff_zero] at hI_00 hI_01 hI_02 hI_10 hI_11 hI_12 hI_20 hI_21 hI_22 hII_00 hII_01 hII_02 hII_10 hII_11 hII_12 hII_20 hII_21 hII_22 hH_00 hH_01 hH_02 hH_10 hH_11 hH_12 hH_20 hH_21 hH_22 hV_00 hV_01 hV_02 hV_10 hV_11 hV_12 hV_20 hV_21 hV_22 hB_00 hB_01 hB_02 hB_10 hB_11 hB_12 hB_20 hB_21 hB_22
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
hI_00 :
  ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_01 :
  ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_02 :
  ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_10 :
  ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_11 :
  ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_12 :
  ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_20 :
  ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_21 :
  ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hI_22 :
  ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
          else 0) +
          ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) *
                (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
            else 0) +
            ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0) +
              if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
              else 0)) +
        if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
      ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum *
            (List.map (fun x => Polynomial.C (if ↑x = 0 then 1 else 0) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum
        else 0) +
        (((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum
                      else 0) +
                    ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0) +
                        -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum
                          else 0)) +
                  -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                            (List.finRange n_wit)).sum
                    else 0) +
                ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                        (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                        else 0) +
                      ((-if
                              (fun₀ | Vars.γ => 1) +
                                  ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0) +
                        -if
                              (fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) =
                                (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                  (List.finRange n_wit)).sum *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                          else 0)) +
                  -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
                    else 0) +
              ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                      (List.map
                            (fun x =>
                              Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum *
                        Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                    else 0) +
                  ((-if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                      else 0) +
                    ((-if
                            (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                              (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0) +
                      -if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                                (List.finRange n_wit)).sum *
                            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                        else 0)) +
                -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                    (List.map
                          (fun x =>
                            Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
                  else 0) +
            ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                  -((List.map
                          (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum *
                      (List.map
                          (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)
                else 0) +
                ((-if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                      Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                        (List.map
                            (fun x =>
                              Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum
                    else 0) +
                  ((-if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                        Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0) +
                    -if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                              (List.finRange n_wit)).sum *
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum
                      else 0)) +
              if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                -((List.map
                        (fun x =>
                          Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum *
                    (List.map
                        (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)
              else 0) +
          ((if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum *
                  (List.map
                      (fun x =>
                        -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                            Polynomial.X ^ ↑x))
                      (List.finRange n_var)).sum
              else 0) +
              ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                  Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                    (List.map
                        (fun x =>
                          -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                              Polynomial.X ^ ↑x))
                        (List.finRange n_var)).sum
                else 0) +
                ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                    Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0) +
                  if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                    (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum *
                      (List.map
                          (fun x =>
                            -(Polynomial.C
                                  (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                                Polynomial.X ^ ↑x))
                          (List.finRange n_var)).sum
                  else 0)) +
            if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
              (List.map
                    (fun x =>
                      Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                        Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x =>
                      -(Polynomial.C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum *
                          Polynomial.X ^ ↑x))
                    (List.finRange n_var)).sum
            else 0))) =
    0
hII_00 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_01 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_02 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_10 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_11 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_12 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_20 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_21 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hII_22 :
  (if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum
        else 0) +
        ((if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
          else 0) +
          ((if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
            else 0) +
            if ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum
            else 0)) +
      ((((-if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                    (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                    (List.finRange n_wit)).sum
              else 0) +
            -if
                  (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1)) =
                    (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
                Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
              else 0) +
          -if
                (fun₀ | Vars.γ => 1) + ((fun₀ | Vars.β => 1) + fun₀ | Vars.β => 1) =
                  (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
            else 0) +
        -if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum
          else 0) =
    0
hH_00 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_01 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_02 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_10 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_11 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_12 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_20 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_21 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hH_22 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_00 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_01 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_02 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_10 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_11 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_12 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_20 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_21 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hV_22 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_00 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_01 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_02 :
  (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 0) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_10 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_11 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_12 :
  (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 1) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_20 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 0 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_21 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 1 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0))
hB_22 :
  (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 0)) =
    (if 0 = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
        (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * Polynomial.X ^ ↑x)
            (List.finRange n_var)).sum
      else 0) +
      ((if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
          Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)
        else 0) +
        ((if ((fun₀ | Vars.γ => 1) + fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ)
          else 0) +
          if (fun₀ | Vars.β => 1) = (fun₀ | Vars.β => 2) + fun₀ | Vars.γ => 2 then
            (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                (List.finRange n_wit)).sum
          else 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