Theorem: Groth16TypeI.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
r : Fin n_wit → F
stmt : Groth16TypeI.Stmt
prover : Prover F Groth16TypeI
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeI.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeI.SRSElementValue_G1 SRS_elem)
                          Groth16TypeI.ListSRSElements_G1).sum)
                  Groth16TypeI.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeI.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeI.SRSElementValue_G1 SRS_elem)
                  Groth16TypeI.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeI.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeI.SRSElementValue_G2 SRS_elem)
                          Groth16TypeI.ListSRSElements_G2).sum)
                  Groth16TypeI.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeI.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeI.SRSElementValue_G2 SRS_elem)
                  Groth16TypeI.ListSRSElements_G2).sum))
        (Groth16TypeI.ListPairings ())).sum =
    0
eqnA :
  (List.map (fun SRS_elem => C (prover.1 Proof_Idx.A SRS_elem) * Groth16TypeI.SRSElementValue_G1 SRS_elem)
        Groth16TypeI.ListSRSElements_G1).sum =
    (List.map (fun SRS_elem => C (prover.2 Proof_Idx.A SRS_elem) * Groth16TypeI.SRSElementValue_G2 SRS_elem)
        Groth16TypeI.ListSRSElements_G2).sum
eqnB :
  (List.map (fun SRS_elem => C (prover.1 Proof_Idx.B SRS_elem) * Groth16TypeI.SRSElementValue_G1 SRS_elem)
        Groth16TypeI.ListSRSElements_G1).sum =
    (List.map (fun SRS_elem => C (prover.2 Proof_Idx.B SRS_elem) * Groth16TypeI.SRSElementValue_G2 SRS_elem)
        Groth16TypeI.ListSRSElements_G2).sum
eqnC :
  (List.map (fun SRS_elem => C (prover.1 Proof_Idx.C SRS_elem) * Groth16TypeI.SRSElementValue_G1 SRS_elem)
        Groth16TypeI.ListSRSElements_G1).sum =
    (List.map (fun SRS_elem => C (prover.2 Proof_Idx.C SRS_elem) * Groth16TypeI.SRSElementValue_G2 SRS_elem)
        Groth16TypeI.ListSRSElements_G2).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.C (SRS_Elements_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum
Tactic:
simp_rw [Groth16TypeI] at eqn eqnA eqnB eqnC
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
r : Fin n_wit → F
stmt : Groth16TypeI.Stmt
prover : Prover F Groth16TypeI
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match pf_elem with
                          | Proof_Idx.A => 1
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 0
                        | PairingsIdx.αβ =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 0
                        | PairingsIdx.stmtγ =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 0
                        | PairingsIdx.cδ =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 1) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                              | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                              | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                              | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                              | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                              | SRS_Elements_Idx.x_pow_times_t i =>
                                X Vars_γ * X Vars_x ^ ↑i *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
                              | SRS_Elements_Idx.y i =>
                                X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_Idx.q i =>
                                X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                          ([SRS_Elements_Idx.α] ++ [SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.δ] ++
                                  List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var) ++
                                List.map (fun i => SRS_Elements_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
                              List.map (fun i => SRS_Elements_Idx.y i) (List.finRange n_stmt) ++
                            List.map (fun i => SRS_Elements_Idx.q i) (List.finRange n_wit))).sum)
                  [Proof_Idx.A, Proof_Idx.B, Proof_Idx.C]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match SRS_elem with
                          | SRS_Elements_Idx.α => 0
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | SRS_Elements_Idx.x_pow_times_t a => 0
                          | SRS_Elements_Idx.y a => 0
                          | SRS_Elements_Idx.q a => 0
                        | PairingsIdx.αβ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.α => 1
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | SRS_Elements_Idx.x_pow_times_t a => 0
                          | SRS_Elements_Idx.y a => 0
                          | SRS_Elements_Idx.q a => 0
                        | PairingsIdx.stmtγ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.α => 0
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | SRS_Elements_Idx.x_pow_times_t a => 0
                          | SRS_Elements_Idx.y i => stmt i
                          | SRS_Elements_Idx.q a => 0
                        | PairingsIdx.cδ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.α => 0
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | SRS_Elements_Idx.x_pow_times_t a => 0
                          | SRS_Elements_Idx.y a => 0
                          | SRS_Elements_Idx.q a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                      | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                      | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                      | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                      | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                      | SRS_Elements_Idx.x_pow_times_t i =>
                        X Vars_γ * X Vars_x ^ ↑i *
                          (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
                      | SRS_Elements_Idx.y i =>
                        X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_Idx.q i =>
                        X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                  ([SRS_Elements_Idx.α] ++ [SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.δ] ++
                          List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var) ++
                        List.map (fun i => SRS_Elements_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
                      List.map (fun i => SRS_Elements_Idx.y i) (List.finRange n_stmt) ++
                    List.map (fun i => SRS_Elements_Idx.q i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => -1
                          | Proof_Idx.C => 0
                        | PairingsIdx.αβ =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 0
                        | PairingsIdx.stmtγ =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 0
                        | PairingsIdx.cδ =>
                          match pf_elem with
                          | Proof_Idx.A => 0
                          | Proof_Idx.B => 0
                          | Proof_Idx.C => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                              | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                              | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                              | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                              | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                              | SRS_Elements_Idx.x_pow_times_t i =>
                                X Vars_γ * X Vars_x ^ ↑i *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
                              | SRS_Elements_Idx.y i =>
                                X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_Idx.q i =>
                                X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                          ([SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.γ] ++ [SRS_Elements_Idx.δ] ++
                            List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var))).sum)
                  [Proof_Idx.A, Proof_Idx.B, Proof_Idx.C]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match SRS_elem with
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | x => 0
                        | PairingsIdx.αβ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.β => 1
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | x => 0
                        | PairingsIdx.stmtγ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 1
                          | SRS_Elements_Idx.δ => 0
                          | SRS_Elements_Idx.x_pow a => 0
                          | x => 0
                        | PairingsIdx.cδ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.β => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.δ => 1
                          | SRS_Elements_Idx.x_pow a => 0
                          | x => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                      | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                      | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                      | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                      | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                      | SRS_Elements_Idx.x_pow_times_t i =>
                        X Vars_γ * X Vars_x ^ ↑i *
                          (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
                      | SRS_Elements_Idx.y i =>
                        X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_Idx.q i =>
                        X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                  ([SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.γ] ++ [SRS_Elements_Idx.δ] ++
                    List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var))).sum))
        [PairingsIdx.ab, PairingsIdx.αβ, PairingsIdx.stmtγ, PairingsIdx.cδ]).sum =
    0
eqnA :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.A SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
            | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
            | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
            | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
            | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
            | SRS_Elements_Idx.x_pow_times_t i =>
              X Vars_γ * X Vars_x ^ ↑i *
                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
            | SRS_Elements_Idx.y i =>
              X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                  X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
            | SRS_Elements_Idx.q i =>
              X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                  X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
        ([SRS_Elements_Idx.α] ++ [SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.δ] ++
                List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var) ++
              List.map (fun i => SRS_Elements_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
            List.map (fun i => SRS_Elements_Idx.y i) (List.finRange n_stmt) ++
          List.map (fun i => SRS_Elements_Idx.q i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.A SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
            | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
            | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
            | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
            | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
            | SRS_Elements_Idx.x_pow_times_t i =>
              X Vars_γ * X Vars_x ^ ↑i *
                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
            | SRS_Elements_Idx.y i =>
              X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                  X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
            | SRS_Elements_Idx.q i =>
              X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                  X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
        ([SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.γ] ++ [SRS_Elements_Idx.δ] ++
          List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var))).sum
eqnB :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.B SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
            | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
            | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
            | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
            | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
            | SRS_Elements_Idx.x_pow_times_t i =>
              X Vars_γ * X Vars_x ^ ↑i *
                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
            | SRS_Elements_Idx.y i =>
              X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                  X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
            | SRS_Elements_Idx.q i =>
              X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                  X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
        ([SRS_Elements_Idx.α] ++ [SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.δ] ++
                List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var) ++
              List.map (fun i => SRS_Elements_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
            List.map (fun i => SRS_Elements_Idx.y i) (List.finRange n_stmt) ++
          List.map (fun i => SRS_Elements_Idx.q 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.α => X Vars_γ * X Vars_δ * X Vars_α
            | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
            | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
            | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
            | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
            | SRS_Elements_Idx.x_pow_times_t i =>
              X Vars_γ * X Vars_x ^ ↑i *
                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
            | SRS_Elements_Idx.y i =>
              X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                  X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
            | SRS_Elements_Idx.q i =>
              X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                  X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
        ([SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.γ] ++ [SRS_Elements_Idx.δ] ++
          List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var))).sum
eqnC :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.C SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
            | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
            | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
            | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
            | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
            | SRS_Elements_Idx.x_pow_times_t i =>
              X Vars_γ * X Vars_x ^ ↑i *
                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
            | SRS_Elements_Idx.y i =>
              X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                  X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
            | SRS_Elements_Idx.q i =>
              X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                  X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
        ([SRS_Elements_Idx.α] ++ [SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.δ] ++
                List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var) ++
              List.map (fun i => SRS_Elements_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
            List.map (fun i => SRS_Elements_Idx.y i) (List.finRange n_stmt) ++
          List.map (fun i => SRS_Elements_Idx.q i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.C SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
            | SRS_Elements_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
            | SRS_Elements_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
            | SRS_Elements_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
            | SRS_Elements_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
            | SRS_Elements_Idx.x_pow_times_t i =>
              X Vars_γ * X Vars_x ^ ↑i *
                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
            | SRS_Elements_Idx.y i =>
              X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                  X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
            | SRS_Elements_Idx.q i =>
              X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                  X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
        ([SRS_Elements_Idx.β] ++ [SRS_Elements_Idx.γ] ++ [SRS_Elements_Idx.δ] ++
          List.map (fun i => SRS_Elements_Idx.x_pow i) (List.finRange n_var))).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.C (SRS_Elements_Idx.q i)) * u_wit i)
            (List.finRange n_wit)).sum) *
      ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * v_wit i)
            (List.finRange n_wit)).sum) =
    (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x => Polynomial.C (prover.1 Proof_Idx.C (SRS_Elements_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum