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
eqnI :
  (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)))) =
    0
eqnII :
  (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) =
    0
eqnH :
  (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)) =
    (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))
eqnV :
  (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)) =
    (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))
eqnB :
  (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)) =
    (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))
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
⊢ ((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:
have hI_02 := congr_arg (coeff (Finsupp.single Vars.β 0 + Finsupp.single Vars.γ 2)) eqnI
State after:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnI :
  (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)))) =
    0
eqnII :
  (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) =
    0
eqnH :
  (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)) =
    (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))
eqnV :
  (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)) =
    (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))
eqnB :
  (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)) =
    (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))
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
⊢ ((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