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
h0021 :
  -((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) =
    0
h0022 :
  -((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)) =
    0
h0112 :
  -((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)) =
    0
h0121 :
  -((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))) =
    0
h0122 :
  -(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) =
    0
h0212 :
  -((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)) =
    0
h0221 :
  -((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)) =
    0
h0222 :
  -(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)) =
    0
h1022 :
  -(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) =
    0
h1112 :
  -((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)) =
    0
h1121 :
  -((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)) =
    0
h1122 :
  -(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 =
    0
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
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0
Tactic:
replace h0021 := congr_arg ((Ideal.Quotient.mk (Ideal.span {t}))) h0021
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
h0022 :
  -((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)) =
    0
h0112 :
  -((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)) =
    0
h0121 :
  -((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))) =
    0
h0122 :
  -(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) =
    0
h0212 :
  -((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)) =
    0
h0221 :
  -((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)) =
    0
h0222 :
  -(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)) =
    0
h1022 :
  -(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) =
    0
h1112 :
  -((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)) =
    0
h1121 :
  -((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)) =
    0
h1122 :
  -(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 =
    0
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
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0