Theorem: BabySNARK.is_sound
Proof state before:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnI :
  C
            ((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) +
          (X Vars.β *
              C
                (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) +
            (X Vars.γ *
                (X Vars.β *
                  C
                    (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)) +
              X Vars.β *
                C
                  ((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))) +
        C
          ((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) +
      (C
          ((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) +
        (-(X Vars.β *
                        C
                          ((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)) +
                    (-(X Vars.β *
                          (X Vars.β *
                            C
                              (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))) +
                      (-(X Vars.γ *
                            (X Vars.β *
                              (X Vars.β *
                                C
                                  (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)))) +
                        -(X Vars.β *
                            (X Vars.β *
                              C
                                ((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))))) +
                  -(X Vars.β *
                      C
                        ((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)) +
                (-(X Vars.γ *
                        (X Vars.β *
                          C
                            ((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)))) +
                    (-(X Vars.β *
                          (X Vars.γ *
                            (X Vars.β *
                              C
                                (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                                  Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))))) +
                      (-(X Vars.γ *
                            (X Vars.β *
                              (X Vars.γ *
                                (X Vars.β *
                                  C
                                    (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                      Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))))) +
                        -(X Vars.β *
                            (X Vars.γ *
                              (X Vars.β *
                                C
                                  ((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))))))) +
                  -(X Vars.γ *
                      (X Vars.β *
                        C
                          ((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))))) +
              (-(X Vars.β *
                      C
                        ((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))) +
                  (-(X Vars.β *
                        (X Vars.β *
                          C
                            (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) *
                              Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)))) +
                    (-(X Vars.γ *
                          (X Vars.β *
                            (X Vars.β *
                              C
                                (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) *
                                  Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))))) +
                      -(X Vars.β *
                          (X Vars.β *
                            C
                              ((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)))))) +
                -(X Vars.β *
                    C
                      ((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)))) +
            (C
                  (-((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)) +
                (-(X Vars.β *
                      C
                        (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)) +
                  (-(X Vars.γ *
                        (X Vars.β *
                          C
                            (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))) +
                    -(X Vars.β *
                        C
                          ((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)))) +
              C
                (-((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))) +
          (C
                ((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) +
              (X Vars.β *
                  C
                    (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) +
                (X Vars.γ *
                    (X Vars.β *
                      C
                        (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)) +
                  X Vars.β *
                    C
                      ((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))) +
            C
              ((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 :
  X Vars.β *
          C
            (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 +
        (X Vars.β * (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ))) +
          (X Vars.γ * (X Vars.β * (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ)))) +
            X Vars.β *
              (X Vars.β *
                C
                  (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                      (List.finRange n_wit)).sum))) +
      (-(X Vars.γ *
                (X Vars.β *
                  (X Vars.β *
                    C
                      (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                          (List.finRange n_wit)).sum))) +
            -(X Vars.γ *
                (X Vars.β * (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ)))))) +
          -(X Vars.γ * (X Vars.β * (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ))))) +
        -(X Vars.γ *
            (X Vars.β *
              C
                (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 :
  C
        (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 +
      (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ)) +
        (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ))) +
          X Vars.β *
            C
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)) =
    C
        (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 +
      (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ)) +
        (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ))) +
          X Vars.β *
            C
              (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 :
  C
        (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 +
      (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ)) +
        (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ))) +
          X Vars.β *
            C
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)) =
    C
        (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 +
      (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ)) +
        (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ))) +
          X Vars.β *
            C
              (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 :
  C
        (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 +
      (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ)) +
        (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ))) +
          X Vars.β *
            C
              (List.map (fun x => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum)) =
    C
        (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 +
      (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ)) +
        (X Vars.γ * (X Vars.β * C (Polynomial.C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ))) +
          X Vars.β *
            C
              (List.map (fun x => Polynomial.C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * u_wit x)
                  (List.finRange n_wit)).sum))
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
          (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu i)) * u_wit i)
              (List.finRange n_wit)).sum) ^
        2 -
      1 =
    t *
      (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu i)) * u_wit i)
          (List.finRange n_wit)).sum
Tactic:
simp only [X, C_apply, monomial_mul, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqnI eqnII eqnH eqnV eqnB
Proof state after:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnI :
  (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))
⊢ ((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