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