Theorem: Groth16TypeIII.is_sound_quotient_version
Proof state before:
case 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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit : Polynomial F ⧸ Ideal.span {t}
h0021 : sum_A_w_wit * sum_B_x = 0
sum_A_v_wit : Polynomial F ⧸ Ideal.span {t}
h1121 : sum_A_v_wit * B_1 = 0
sum_A_u_wit : Polynomial F ⧸ Ideal.span {t}
h0221 : sum_A_u_wit * B_1 = 0
h0121 : -(sum_A_u_wit * sum_B_x) + -(sum_A_w_wit * B_1) = 0
sum_A_w_stmt : Polynomial F ⧸ Ideal.span {t}
h0012 : sum_A_w_stmt * sum_B_x = 0
sum_A_v_stmt : Polynomial F ⧸ Ideal.span {t}
h1112 : sum_A_v_stmt * B_1 = 0
sum_A_u_stmt : Polynomial F ⧸ Ideal.span {t}
h0212 : sum_A_u_stmt * B_1 = 0
h0112 : -(sum_A_u_stmt * sum_B_x) + -(sum_A_w_stmt * B_1) = 0
sum_A_x A_3 A_2 : Polynomial F ⧸ Ideal.span {t}
h0222 : A_2 * B_1 = 0
A_1 : Polynomial F ⧸ Ideal.span {t}
h1122 : -(A_1 * B_1) + 1 = 0
sum_w_stmt : Polynomial F ⧸ Ideal.span {t}
h0022 : -(sum_A_x * sum_B_x) + -(sum_A_w_wit * B_3) + -(sum_A_w_stmt * B_2) + (sum_w_stmt + sum_C_w_wit) = 0
sum_v_stmt : Polynomial F ⧸ Ideal.span {t}
h1022 : -(A_1 * sum_B_x) + -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) + (sum_v_stmt + sum_C_v_wit) = 0
sum_u_stmt : Polynomial F ⧸ Ideal.span {t}
h0122 :
  -(A_2 * sum_B_x) + -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) + -(sum_A_x * B_1) + (sum_u_stmt + sum_C_u_wit) = 0
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.hq
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
h0012 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
              (List.finRange n_stmt)).sum *
          (List.map (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0021 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map
                (fun x =>
                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                    (Polynomial.X ^ ↑x * t))
                (List.finRange (n_var - 1))).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum) +
        -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                (List.finRange n_wit)).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0022 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum) +
            (-((List.map
                      (fun x =>
                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                          (Polynomial.X ^ ↑x * t))
                      (List.finRange (n_var - 1))).sum *
                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
              -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                      (List.finRange n_wit)).sum *
                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))) +
          -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                  (List.finRange n_stmt)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
        ((List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum +
          ((List.map
                (fun x =>
                  Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                    (Polynomial.X ^ ↑x * t))
                (List.finRange (n_var - 1))).sum +
            (List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * w_wit x)
                (List.finRange n_wit)).sum))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0112 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                (List.finRange n_stmt)).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum) +
        -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                (List.finRange n_stmt)).sum *
            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0121 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
                (List.finRange n_wit)).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum) +
        (-((List.map
                  (fun x =>
                    Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                      (Polynomial.X ^ ↑x * t))
                  (List.finRange (n_var - 1))).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
          -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                  (List.finRange n_wit)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0122 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                  (List.map
                      (fun x =>
                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum) +
              -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
            -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                    (List.finRange n_stmt)).sum *
                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
          -((List.map
                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
        ((List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum +
          (List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * u_wit x)
              (List.finRange n_wit)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0212 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
              (List.finRange n_stmt)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0221 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
              (List.finRange n_wit)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0222 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1022 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                (List.map
                    (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum) +
            -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * v_wit x)
                    (List.finRange n_wit)).sum *
                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
          -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * v_stmt x)
                  (List.finRange n_stmt)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
        ((List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum +
          (List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * v_wit x)
              (List.finRange n_wit)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1112 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * v_stmt x)
              (List.finRange n_stmt)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1121 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * v_wit x)
              (List.finRange n_wit)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1122 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
        1) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
⊢ t.Monic
Tactic:
integral_domain_tactic
Proof state after:
case intro.inl.inl.inl.inl.inl.inl.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : -(A_1 * B_1) = -1
h0022 : -(sum_A_x * sum_B_x) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(A_1 * sum_B_x) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_x * B_1) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0222 : A_2 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inl.inl.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_x * sum_B_x) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(A_1 * sum_B_x) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(A_2 * sum_B_x) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0222 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inl.inl.inl.inr.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_x * sum_B_x) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(A_1 * sum_B_x) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(A_2 * sum_B_x) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h1112 : sum_A_v_stmt = 0
h0212 : B_1 = 0
h0112 : sum_A_u_stmt = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inl.inl.inl.inr.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h1112 : sum_A_v_stmt = 0
h0212 : B_1 = 0
h0112 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inl.inl.inl.inl.inr.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_x * sum_B_x) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(A_1 * sum_B_x) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(A_2 * sum_B_x) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h1112 : B_1 = 0
h0112 : sum_A_u_stmt = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inl.inl.inr.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h1112 : B_1 = 0
h0112 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inl.inl.inl.inr.inl.inl.inl.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : -(A_1 * B_1) = -1
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : -(sum_A_x * B_1) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0112 : sum_A_w_stmt = 0
h0222 : A_2 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inl.inl.inl.inr.inl.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : sum_u_stmt = -sum_C_u_wit
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0112 : sum_A_w_stmt = 0
h0222 : B_1 = 0
⊢ (-sum_C_u_wit + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inl.inl.inl.inr.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : sum_u_stmt = -sum_C_u_wit
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0112 : B_1 = 0
⊢ (-sum_C_u_wit + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inl.inr.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
h1112 : sum_A_v_stmt = 0
h0212 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inl.inr.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
h1112 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inr.inl.inl.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_x * sum_B_x) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(A_1 * sum_B_x) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(A_2 * sum_B_x) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : B_1 = 0
h0121 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h0112 : sum_A_u_stmt = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inr.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : B_1 = 0
h0121 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h0112 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inl.inl.inr.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : B_1 = 0
h0121 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inl.inr.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : sum_A_v_wit = 0
h0221 : B_1 = 0
h0121 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inr.inl.inl.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_x * sum_B_x) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(A_1 * sum_B_x) + -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(A_2 * sum_B_x) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : B_1 = 0
h0121 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h0112 : sum_A_u_stmt = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inr.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : B_1 = 0
h0121 : sum_A_u_wit = 0
h0012 : sum_A_w_stmt = 0
h0112 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inl.inr.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : B_1 = 0
h0121 : sum_A_u_wit = 0
h0012 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inl.inr.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_A_w_wit = 0
h1121 : B_1 = 0
h0121 : sum_B_x = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inr.inl.inl.inl.inl.inl.inl.inl
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : -(A_1 * B_1) = -1
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : -(sum_A_x * B_1) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0121 : sum_A_w_wit = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0112 : sum_A_w_stmt = 0
h0222 : A_2 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inr.inl.inl.inl.inl.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : sum_w_stmt = -sum_C_w_wit
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : sum_u_stmt = -sum_C_u_wit
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0121 : sum_A_w_wit = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0112 : sum_A_w_stmt = 0
h0222 : B_1 = 0
⊢ (-sum_C_u_wit + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (-sum_C_w_wit + sum_C_w_wit) = 0

case intro.inr.inl.inl.inl.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : sum_u_stmt = -sum_C_u_wit
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0121 : sum_A_w_wit = 0
h1112 : sum_A_v_stmt = 0
h0212 : sum_A_u_stmt = 0
h0112 : B_1 = 0
⊢ (-sum_C_u_wit + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inr.inl.inl.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : sum_v_stmt = -sum_C_v_wit
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0121 : sum_A_w_wit = 0
h1112 : sum_A_v_stmt = 0
h0212 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (-sum_C_v_wit + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inr.inl.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0121 : sum_A_w_wit = 0
h1112 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inr.inl.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_wit * B_3) + -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : sum_A_u_wit = 0
h0121 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inr.inl.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_wit * B_3) + -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : sum_A_v_wit = 0
h0221 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.inr.inr
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
sum_C_x_t sum_C_w_wit sum_C_v_wit sum_C_u_wit : Polynomial F ⧸ Ideal.span {t}
C_3 C_2 C_1 : Polynomial F
sum_B_x B_3 B_2 B_1 sum_A_x_t sum_A_w_wit sum_A_v_wit sum_A_u_wit sum_A_w_stmt sum_A_v_stmt sum_A_u_stmt sum_A_x A_3 A_2 A_1 sum_w_stmt sum_v_stmt sum_u_stmt :
  Polynomial F ⧸ Ideal.span {t}
h1122 : 1 = 0
h0022 : -(sum_A_w_wit * B_3) + -(sum_A_w_stmt * B_2) = -(sum_w_stmt + sum_C_w_wit)
h1022 : -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) = -(sum_v_stmt + sum_C_v_wit)
h0122 : -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) = -(sum_u_stmt + sum_C_u_wit)
this : NoZeroDivisors (Polynomial F ⧸ Ideal.span {t})
h0021 : sum_B_x = 0
h1121 : B_1 = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) - (sum_w_stmt + sum_C_w_wit) = 0

case intro.hq
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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
h0012 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
              (List.finRange n_stmt)).sum *
          (List.map (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
              (List.finRange n_var)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0021 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map
                (fun x =>
                  Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                    (Polynomial.X ^ ↑x * t))
                (List.finRange (n_var - 1))).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum) +
        -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                (List.finRange n_wit)).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0022 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum *
                (List.map
                    (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum) +
            (-((List.map
                      (fun x =>
                        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                          (Polynomial.X ^ ↑x * t))
                      (List.finRange (n_var - 1))).sum *
                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
              -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                      (List.finRange n_wit)).sum *
                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))) +
          -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                  (List.finRange n_stmt)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
        ((List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum +
          ((List.map
                (fun x =>
                  Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                    (Polynomial.X ^ ↑x * t))
                (List.finRange (n_var - 1))).sum +
            (List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * w_wit x)
                (List.finRange n_wit)).sum))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0112 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                (List.finRange n_stmt)).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum) +
        -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                (List.finRange n_stmt)).sum *
            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0121 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
                (List.finRange n_wit)).sum *
            (List.map
                (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                (List.finRange n_var)).sum) +
        (-((List.map
                  (fun x =>
                    Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                      (Polynomial.X ^ ↑x * t))
                  (List.finRange (n_var - 1))).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
          -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                  (List.finRange n_wit)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0122 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                  (List.map
                      (fun x =>
                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                      (List.finRange n_var)).sum) +
              -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
                      (List.finRange n_wit)).sum *
                  Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
            -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                    (List.finRange n_stmt)).sum *
                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
          -((List.map
                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                  (List.finRange n_var)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
        ((List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum +
          (List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * u_wit x)
              (List.finRange n_wit)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0212 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
              (List.finRange n_stmt)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0221 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
              (List.finRange n_wit)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h0222 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1022 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                (List.map
                    (fun x => Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                    (List.finRange n_var)).sum) +
            -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * v_wit x)
                    (List.finRange n_wit)).sum *
                Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)) +
          -((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * v_stmt x)
                  (List.finRange n_stmt)).sum *
              Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)) +
        ((List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum +
          (List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * v_wit x)
              (List.finRange n_wit)).sum)) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1112 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * v_stmt x)
              (List.finRange n_stmt)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1121 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-((List.map (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * v_wit x)
              (List.finRange n_wit)).sum *
          Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
h1122 :
  (Ideal.Quotient.mk (Ideal.span {t}))
      (-(Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
            Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)) +
        1) =
    (Ideal.Quotient.mk (Ideal.span {t})) 0
⊢ t.Monic