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 :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match pf_elem with
                          | Proof_Idx.H => 1
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.gg =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.vv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 1
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => t.coeff ↑i
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.gg =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => if ↑i = 0 then 1 else 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.vv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i =>
                            (List.map (fun j => stmt j * (u_stmt j).coeff ↑i) (List.finRange n_stmt)).sum
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.gg =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.vv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => -1
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => t.coeff ↑i
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.gg =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => if ↑i = 0 then 1 else 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.vv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i =>
                            -(List.map (fun j => stmt j * (u_stmt j).coeff ↑i) (List.finRange n_stmt)).sum
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum))
        [PairingsI_Idx.ht, PairingsI_Idx.gg, PairingsI_Idx.vv]).sum =
    0
eqnII :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 1
                        | PairingsII_Idx.γβv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsII_Idx.γβv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 1
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsII_Idx.γβv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => -1
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 1
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsII_Idx.γβv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum))
        [PairingsII_Idx.bγ, PairingsII_Idx.γβv]).sum =
    0
eqnH :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.H SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.H SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum
eqnV :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.V SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.V SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum
eqnB :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.B SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.B SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (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 [monomial_zero', List.singleton_append, List.cons_append, List.append_assoc,
  List.map_cons, Sum.elim_inl, Sum.elim_inr, List.map_append, List.map_map, List.sum_cons,
  List.sum_append, List.map_nil, List.sum_nil, add_zero, Sum.elim_lam_const_lam_const, map_one,
  one_mul, map_zero, zero_mul, map_neg, neg_mul, neg_add_rev, zero_add, mul_zero,
  Function.comp, List.sum_map_zero] at eqnI eqnII eqnH eqnV eqnB
Proof state after:
case intro.intro.intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnI :
  ((List.map (fun x => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
            (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) * X Vars_β +
              (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
                (List.map
                    (fun x =>
                      C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) *
                        (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                    (List.finRange n_wit)).sum)) +
          (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum) *
        (List.map (fun x => C (t.coeff ↑x) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      ((List.map (fun x => C (if ↑x = 0 then 1 else 0) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum *
          (List.map (fun x => C (if ↑x = 0 then 1 else 0) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
        ((List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                  (List.finRange n_var)).sum +
              (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β +
                (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
                  (List.map
                      (fun x =>
                        C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                          (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                      (List.finRange n_wit)).sum)) +
            (List.map
                (fun x =>
                  C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x)
                (List.finRange n_var)).sum) *
          (-(List.map
                        (fun x =>
                          C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                            (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                        (List.finRange n_wit)).sum +
                  -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)) +
                -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β) +
              -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                    (List.finRange n_var)).sum +
            (List.map
                (fun x =>
                  -(C (List.map (fun j => stmt j * (u_stmt j).coeff ↑x) (List.finRange n_stmt)).sum * X Vars_τ ^ ↑x))
                (List.finRange n_var)).sum)) =
    0
eqnII :
  ((List.map (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
          (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ) * X Vars_β +
            (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
              (List.map
                  (fun x =>
                    C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) *
                      (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                  (List.finRange n_wit)).sum))) *
        X Vars_β +
      X Vars_γ * X Vars_β *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) *
                        (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
                    (List.finRange n_wit)).sum +
              -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β)) +
            -(C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β) +
          -(List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
                (List.finRange n_var)).sum) =
    0
eqnH :
  (List.map (fun x => C (prover.1 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.1 Proof_Idx.H SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.H (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (C (prover.2 Proof_Idx.H SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.2 Proof_Idx.H SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.2 Proof_Idx.H (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum))
eqnV :
  (List.map (fun x => C (prover.1 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.1 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.1 Proof_Idx.V (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.V (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (C (prover.2 Proof_Idx.V SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.2 Proof_Idx.V SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.2 Proof_Idx.V (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum))
eqnB :
  (List.map (fun x => C (prover.1 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x) (List.finRange n_var)).sum +
      (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.1 Proof_Idx.B SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum)) =
    (List.map (fun x => C (prover.2 Proof_Idx.B (SRS_Elements_Idx.τ_pow x)) * X Vars_τ ^ ↑x)
          (List.finRange n_var)).sum +
      (C (prover.2 Proof_Idx.B SRS_Elements_Idx.γ) * X Vars_β +
        (C (prover.2 Proof_Idx.B SRS_Elements_Idx.γβ) * (X Vars_γ * X Vars_β) +
          (List.map
              (fun x =>
                C (prover.2 Proof_Idx.B (SRS_Elements_Idx.βu x)) * (X Vars_β * (to_MvPolynomial_Option Vars) (u_wit x)))
              (List.finRange n_wit)).sum))
⊢ ((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